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      <a class="article-title" href="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/">数据结构</a>
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        <a href="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/" class="archive-article-date">
  	<time datetime="2021-05-04T08:06:14.000Z" itemprop="datePublished"><i class="icon-calendar icon"></i>2021-05-04</time>
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<ul>
<li><a href="#%E6%A0%91">树</a><ul>
<li><a href="#%E7%BB%93%E7%82%B9%E6%95%B0--%E7%BB%93%E7%82%B9%E7%9A%84%E5%BA%A6%E4%B9%8B%E5%92%8C%E5%88%86%E6%94%AF%E6%95%B0--1">结点数 = 结点的度之和(分支数) + 1</a></li>
<li><a href="#%E5%BA%A6%E4%B8%BAm%E7%9A%84%E6%A0%91%E4%B8%AD%E7%AC%ACi%E5%B1%82%E4%B8%8A%E8%87%B3%E5%A4%9A%E6%9C%89m%5Ei-1%E4%B8%AA%E7%BB%93%E7%82%B9i%E2%89%A51">度为m的树中第i层上至多有m^(i-1)个结点(i≥1)</a></li>
<li><a href="#%E9%AB%98%E5%BA%A6%E4%B8%BAh%E7%9A%84m%E6%AC%A1%E6%A0%91%E8%87%B3%E5%A4%9A%E6%9C%89m%5Eh-1m-1%E4%B8%AA%E7%BB%93%E7%82%B9%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%E6%B1%82%E5%92%8C">高度为h的m次树至多有(m^h-1)/(m-1)个结点(等比数列求和)</a></li>
<li><a href="#%E5%85%B7%E6%9C%89n%E4%B8%AA%E7%BB%93%E7%82%B9%E7%9A%84m%E6%AC%A1%E6%A0%91%E7%9A%84%E6%9C%80%E5%B0%8F%E9%AB%98%E5%BA%A6%E4%B8%BAlogmnm-11%E5%90%91%E4%B8%8A%E5%8F%96%E6%95%B4">具有n个结点的m次树的最小高度为logm(n(m-1)+1)向上取整</a></li>
<li><a href="#%E6%A0%91%E7%9A%84%E5%AD%98%E5%82%A8%E7%BB%93%E6%9E%84">树的存储结构</a><ul>
<li><a href="#%E5%8F%8C%E4%BA%B2%E5%AD%98%E5%82%A8%E7%BB%93%E6%9E%84">双亲存储结构</a><ul>
<li><a href="#%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0">代码实现</a></li>
</ul>
</li>
<li><a href="#%E5%AD%A9%E5%AD%90%E9%93%BE%E5%AD%98%E5%82%A8%E7%BB%93%E6%9E%84">孩子链存储结构</a><ul>
<li><a href="#%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0-1">代码实现</a></li>
</ul>
</li>
<li><a href="#%E6%B1%82%E6%A0%91%E9%AB%98">求树高</a></li>
</ul>
</li>
<li><a href="#%E5%AD%A9%E5%AD%90%E5%85%84%E5%BC%9F%E9%93%BE%E5%AD%98%E5%82%A8%E7%BB%93%E6%9E%84">孩子兄弟链存储结构</a><ul>
<li><a href="#%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0-2">代码实现</a></li>
<li><a href="#%E6%B1%82%E6%A0%91%E9%AB%98%E5%BA%A6">求树高度</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#%E4%BA%8C%E5%8F%89%E6%A0%91">二叉树</a><ul>
<li><a href="#%E5%AE%9A%E4%B9%89">定义</a></li>
<li><a href="#%E6%BB%A1%E4%BA%8C%E5%8F%89%E6%A0%91">满二叉树</a></li>
<li><a href="#%E5%AE%8C%E5%85%A8%E4%BA%8C%E5%8F%89%E6%A0%91">完全二叉树</a></li>
<li><a href="#%E4%BA%8C%E5%8F%89%E6%A0%91%E6%80%A7%E8%B4%A8">二叉树性质</a><ul>
<li><a href="#%E6%99%AE%E9%80%9A%E4%BA%8C%E5%8F%89%E6%A0%91%E6%80%A7%E8%B4%A8">普通二叉树性质</a></li>
<li><a href="#%E5%AE%8C%E5%85%A8%E4%BA%8C%E5%8F%89%E6%A0%91%E6%80%A7%E8%B4%A8">完全二叉树性质</a></li>
</ul>
</li>
<li><a href="#%E4%BA%8C%E5%8F%89%E6%A0%91%E4%B8%8E%E6%A0%91%E6%A3%AE%E6%9E%97%E4%B9%8B%E9%97%B4%E7%9A%84%E8%BD%AC%E6%8D%A2">二叉树与树、森林之间的转换</a><ul>
<li><a href="#%E6%A3%AE%E6%9E%97%E6%A0%91%E8%BD%AC%E6%8D%A2%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91">森林、树转换为二叉树</a></li>
<li><a href="#%E4%BA%8C%E5%8F%89%E6%A0%91%E8%BF%98%E5%8E%9F%E4%B8%BA%E6%A3%AE%E6%9E%97%E6%A0%91">二叉树还原为森林、树</a></li>
</ul>
</li>
<li><a href="#%E4%BA%8C%E5%8F%89%E6%A0%91%E7%9A%84%E5%85%B3%E7%B3%BB">二叉树的关系</a></li>
<li><a href="#%E4%BA%8C%E5%8F%89%E6%A0%91%E7%9A%84%E5%AD%98%E5%82%A8%E7%BB%93%E6%9E%84">二叉树的存储结构</a><ul>
<li><a href="#%E9%A1%BA%E5%BA%8F%E5%AD%98%E5%82%A8">顺序存储</a><ul>
<li><a href="#%E5%9B%BE%E7%A4%BA">图示</a></li>
<li><a href="#%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0-3">代码实现</a></li>
<li><a href="#%E7%89%B9%E7%82%B9">特点</a></li>
</ul>
</li>
<li><a href="#%E9%93%BE%E5%BC%8F%E5%AD%98%E5%82%A8%E4%BA%8C%E5%8F%89%E9%93%BE">链式存储(二叉链)</a><ul>
<li><a href="#%E5%9B%BE%E7%A4%BA-1">图示</a></li>
<li><a href="#2722-%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0">2.7.2.2. 代码实现</a></li>
<li><a href="#2723-%E7%89%B9%E7%82%B9">2.7.2.3. 特点</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#28-%E4%BA%8C%E5%8F%89%E6%A0%91%E5%9F%BA%E6%9C%AC%E8%BF%90%E7%AE%97">2.8. 二叉树基本运算</a><ul>
<li><a href="#281-%E5%88%9B%E5%BB%BA%E4%BA%8C%E5%8F%89%E6%A0%91">2.8.1. 创建二叉树</a><ul>
<li><a href="#2811-%E7%AE%97%E6%B3%95%E8%AE%BE%E8%AE%A1">2.8.1.1. 算法设计</a></li>
<li><a href="#2812-%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0">2.8.1.2. 代码实现</a></li>
</ul>
</li>
<li><a href="#282-%E9%94%80%E6%AF%81%E4%BA%8C%E5%8F%89%E6%A0%91">2.8.2. 销毁二叉树</a><ul>
<li><a href="#2821-%E7%AE%97%E6%B3%95%E8%AE%BE%E8%AE%A1">2.8.2.1. 算法设计</a></li>
<li><a href="#2822-%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0">2.8.2.2. 代码实现</a></li>
</ul>
</li>
<li><a href="#283-%E6%9F%A5%E6%89%BE%E7%BB%93%E7%82%B9">2.8.3. 查找结点</a><ul>
<li><a href="#2831-%E7%AE%97%E6%B3%95%E8%AE%BE%E8%AE%A1">2.8.3.1. 算法设计</a></li>
<li><a href="#2832-%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0">2.8.3.2. 代码实现</a></li>
</ul>
</li>
<li><a href="#284-%E6%B1%82%E9%AB%98%E5%BA%A6">2.8.4. 求高度</a><ul>
<li><a href="#2841-%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0">2.8.4.1. 代码实现</a></li>
</ul>
</li>
<li><a href="#285-%E8%BE%93%E5%87%BA%E4%BA%8C%E5%8F%89%E6%A0%91">2.8.5. 输出二叉树</a><ul>
<li><a href="#2851-%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0">2.8.5.1. 代码实现</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#29-%E4%BA%8C%E5%8F%89%E6%A0%91%E7%9A%84%E9%81%8D%E5%8E%86">2.9. 二叉树的遍历</a><ul>
<li><a href="#291-%E5%85%88%E5%BA%8F%E9%81%8D%E5%8E%86">2.9.1. 先序遍历</a></li>
<li><a href="#292-%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0">2.9.2. 代码实现</a></li>
<li><a href="#293-%E4%B8%AD%E5%BA%8F%E9%81%8D%E5%8E%86">2.9.3. 中序遍历</a></li>
<li><a href="#294-%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0">2.9.4. 代码实现</a></li>
<li><a href="#295-%E5%90%8E%E5%BA%8F%E9%81%8D%E5%8E%86">2.9.5. 后序遍历</a></li>
<li><a href="#296-%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0">2.9.6. 代码实现</a></li>
</ul>
</li>
<li><a href="#210-%E4%BA%8C%E5%8F%89%E6%A0%91%E7%9A%84%E6%9E%84%E9%80%A0">2.10. 二叉树的构造</a><ul>
<li><a href="#2101-%E6%9D%A1%E4%BB%B6">2.10.1. 条件</a></li>
<li><a href="#2102-%E4%B8%AD%E5%BA%8F--%E5%89%8D%E5%BA%8F">2.10.2. 中序 + 前序</a><ul>
<li><a href="#21021-%E9%80%92%E5%BD%92%E5%8E%9F%E7%90%86">2.10.2.1. 递归原理</a></li>
<li><a href="#21022-%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0">2.10.2.2. 代码实现</a></li>
</ul>
</li>
<li><a href="#2103-%E4%B8%AD%E5%BA%8F--%E5%90%8E%E5%BA%8F">2.10.3. 中序 + 后序</a><ul>
<li><a href="#21031-%E9%80%92%E5%BD%92%E5%8E%9F%E7%90%86">2.10.3.1. 递归原理</a></li>
<li><a href="#21032-%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0">2.10.3.2. 代码实现</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#%E7%BA%BF%E7%B4%A2%E4%BA%8C%E5%8F%89%E6%A0%91">线索二叉树</a><ul>
<li><a href="#%E7%BB%93%E6%9E%84">结构</a><ul>
<li><a href="#%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0-4">代码实现</a></li>
</ul>
</li>
<li><a href="#%E7%BA%BF%E7%B4%A2%E5%8C%96%E4%BA%8C%E5%8F%89%E6%A0%91">线索化二叉树</a><ul>
<li><a href="#%E6%96%B9%E6%B3%95">方法</a></li>
<li><a href="#%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0-5">代码实现</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#%E5%93%88%E5%A4%AB%E6%9B%BC%E6%A0%91">哈夫曼树</a><ul>
<li><a href="#%E8%AE%A1%E7%AE%97wpl">计算WPL</a><ul>
<li><a href="#%E6%A0%B9%E7%BB%93%E7%82%B9%E5%88%B0%E5%90%84%E4%B8%AA%E5%8F%B6%E7%BB%93%E7%82%B9%E7%9A%84%E8%B7%AF%E5%BE%84%E9%95%BF%E5%BA%A6%E4%B8%8E%E7%9B%B8%E5%BA%94%E7%BB%93%E7%82%B9%E6%9D%83%E5%80%BC%E7%9A%84%E4%B9%98%E7%A7%AF%E7%9A%84%E5%92%8C%E5%8F%AB%E5%81%9A%E4%BA%8C%E5%8F%89%E6%A0%91%E7%9A%84%E5%B8%A6%E6%9D%83%E8%B7%AF%E5%BE%84%E9%95%BF%E5%BA%A6wpl">根结点到各个叶结点的路径长度与相应结点权值的乘积的和，叫做二叉树的带权路径长度WPL。</a></li>
</ul>
</li>
<li><a href="#%E6%9E%84%E9%80%A0%E5%93%88%E5%A4%AB%E6%9B%BC%E6%A0%91">构造哈夫曼树</a><ul>
<li><a href="#"></a></li>
</ul>
</li>
<li><a href="#%E5%93%88%E5%A4%AB%E6%9B%BC%E7%BC%96%E7%A0%81">哈夫曼编码</a><ul>
<li><a href="#%E7%89%B9%E7%82%B9-1">特点</a></li>
<li><a href="#%E7%A4%BA%E4%BE%8B">示例</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#%E5%B9%B6%E6%9F%A5%E9%9B%86">并查集</a><ul>
<li><a href="#%E6%9E%84%E9%80%A0">构造</a></li>
<li><a href="#%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84">数据结构</a><ul>
<li><a href="#%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0-6">代码实现</a></li>
</ul>
</li>
<li><a href="#%E5%88%9D%E5%A7%8B%E5%8C%96">初始化</a><ul>
<li><a href="#%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0-7">代码实现</a></li>
</ul>
</li>
<li><a href="#%E6%9F%A5%E6%89%BE%E4%B8%80%E4%B8%AA%E5%85%83%E7%B4%A0%E6%89%80%E5%B1%9E%E7%9A%84%E9%9B%86%E5%90%88">查找一个元素所属的集合</a><ul>
<li><a href="#%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0-8">代码实现</a></li>
</ul>
</li>
<li><a href="#%E4%B8%A4%E4%B8%AA%E5%85%83%E7%B4%A0%E5%90%84%E8%87%AA%E6%89%80%E5%B1%9E%E7%9A%84%E9%9B%86%E5%90%88%E7%9A%84%E5%90%88%E5%B9%B6">两个元素各自所属的集合的合并</a><ul>
<li><a href="#%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0-9">代码实现</a></li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
<li><a href="#%E5%9B%BE">图</a><ul>
<li><a href="#%E5%88%86%E7%B1%BB">分类</a></li>
<li><a href="#%E5%AE%8C%E5%85%A8%E5%9B%BE">完全图</a></li>
<li><a href="#%E5%AD%90%E5%9B%BE">子图</a><ul>
<li><a href="#%E6%A6%82%E5%BF%B5">概念：</a></li>
</ul>
</li>
<li><a href="#%E8%B7%AF%E5%BE%84%E5%92%8C%E8%B7%AF%E5%BE%84%E9%95%BF%E5%BA%A6">路径和路径长度</a><ul>
<li><a href="#%E8%B7%AF%E5%BE%84-%E5%9C%A8%E4%B8%80%E4%B8%AA%E5%9B%BEgve%E4%B8%AD%E4%BB%8E%E9%A1%B6%E7%82%B9i%E5%88%B0%E9%A1%B6%E7%82%B9j%E7%9A%84%E4%B8%80%E6%9D%A1%E8%B7%AF%E5%BE%84ii1i2imj">路径: 在一个图G=(V，E)中，从顶点i到顶点j的一条路径(i，i1，i2，…，im，j)。</a></li>
<li><a href="#%E8%B7%AF%E5%BE%84%E9%95%BF%E5%BA%A6-%E6%98%AF%E6%8C%87%E4%B8%80%E6%9D%A1%E8%B7%AF%E5%BE%84%E4%B8%8A%E7%BB%8F%E8%BF%87%E7%9A%84%E8%BE%B9%E7%9A%84%E6%95%B0%E7%9B%AE">路径长度: 是指一条路径上经过的边的数目</a></li>
<li><a href="#%E7%AE%80%E5%8D%95%E8%B7%AF%E5%BE%84-%E8%8B%A5%E4%B8%80%E6%9D%A1%E8%B7%AF%E5%BE%84%E4%B8%8A%E9%99%A4%E5%BC%80%E5%A7%8B%E7%82%B9%E5%92%8C%E7%BB%93%E6%9D%9F%E7%82%B9%E5%8F%AF%E4%BB%A5%E7%9B%B8%E5%90%8C%E5%A4%96%E5%85%B6%E4%BD%99%E9%A1%B6%E7%82%B9%E5%9D%87%E4%B8%8D%E7%9B%B8%E5%90%8C%E5%88%99%E7%A7%B0%E6%AD%A4%E8%B7%AF%E5%BE%84%E4%B8%BA%E7%AE%80%E5%8D%95%E8%B7%AF%E5%BE%84">简单路径: 若一条路径上除开始点和结束点可以相同外，其余顶点均不相同，则称此路径为简单路径</a></li>
</ul>
</li>
<li><a href="#%E8%BF%9E%E9%80%9A%E8%BF%9E%E9%80%9A%E5%9B%BE%E5%92%8C%E8%BF%9E%E9%80%9A%E5%88%86%E9%87%8F%E6%97%A0%E5%90%91%E5%9B%BE">连通、连通图和连通分量（无向图）</a><ul>
<li><a href="#%E8%BF%9E%E9%80%9A%E8%8B%A5%E4%BB%8E%E9%A1%B6%E7%82%B9i%E5%88%B0%E9%A1%B6%E7%82%B9j%E6%9C%89%E8%B7%AF%E5%BE%84%E5%88%99%E7%A7%B0%E9%A1%B6%E7%82%B9i%E5%92%8Cj%E6%98%AF%E8%BF%9E%E9%80%9A%E7%9A%84">连通：若从顶点i到顶点j有路径，则称顶点i和j是连通的</a></li>
<li><a href="#%E8%BF%9E%E9%80%9A%E5%9B%BE%E8%8B%A5%E5%9B%BE%E4%B8%AD%E4%BB%BB%E6%84%8F%E4%B8%A4%E4%B8%AA%E9%A1%B6%E7%82%B9%E9%83%BD%E8%BF%9E%E9%80%9A%E5%88%99%E7%A7%B0%E4%B8%BA%E8%BF%9E%E9%80%9A%E5%9B%BE%E5%90%A6%E5%88%99%E7%A7%B0%E4%B8%BA%E9%9D%9E%E8%BF%9E%E9%80%9A%E5%9B%BE">连通图：若图中<strong>任意两个顶点都连通</strong>，则称为连通图，否则称为非连通图</a></li>
<li><a href="#%E8%BF%9E%E9%80%9A%E5%88%86%E9%87%8F%E6%97%A0%E5%90%91%E5%9B%BEg%E4%B8%AD%E7%9A%84%E6%9E%81%E5%A4%A7%E8%BF%9E%E9%80%9A%E5%AD%90%E5%9B%BE%E7%A7%B0%E4%B8%BAg%E7%9A%84%E8%BF%9E%E9%80%9A%E5%88%86%E9%87%8F%E6%98%BE%E7%84%B6%E4%BB%BB%E4%BD%95%E8%BF%9E%E9%80%9A%E5%9B%BE%E7%9A%84%E8%BF%9E%E9%80%9A%E5%88%86%E9%87%8F%E5%8F%AA%E6%9C%89%E4%B8%80%E4%B8%AA%E5%8D%B3%E6%9C%AC%E8%BA%AB%E8%80%8C%E9%9D%9E%E8%BF%9E%E9%80%9A%E5%9B%BE%E6%9C%89%E5%A4%9A%E4%B8%AA%E8%BF%9E%E9%80%9A%E5%88%86%E9%87%8F">连通分量：无向图G中的极大连通子图称为G的连通分量。显然，任何连通图的连通分量只有一个，即本身，而非连通图有多个连通分量</a></li>
</ul>
</li>
<li><a href="#%E5%BC%BA%E8%BF%9E%E9%80%9A%E5%9B%BE%E5%92%8C%E5%BC%BA%E8%BF%9E%E9%80%9A%E5%88%86%E9%87%8F%E6%9C%89%E5%90%91%E5%9B%BE">强连通图和强连通分量（有向图）</a><ul>
<li><a href="#%E5%BC%BA%E8%BF%9E%E9%80%9A%E5%9B%BE%E8%8B%A5%E5%9B%BEg%E4%B8%AD%E7%9A%84%E4%BB%BB%E6%84%8F%E4%B8%A4%E4%B8%AA%E9%A1%B6%E7%82%B9i%E5%92%8Cj%E9%83%BD%E8%BF%9E%E9%80%9A%E5%8D%B3%E4%BB%8E%E9%A1%B6%E7%82%B9i%E5%88%B0j%E5%92%8C%E4%BB%8E%E9%A1%B6%E7%82%B9j%E5%88%B0i%E9%83%BD%E5%AD%98%E5%9C%A8%E8%B7%AF%E5%BE%84%E5%88%99%E7%A7%B0%E5%9B%BEg%E6%98%AF%E5%BC%BA%E8%BF%9E%E9%80%9A%E5%9B%BE">强连通图：若图G中的任意两个顶点i和j都连通，即从顶点i到j和从顶点j到i都存在路径，则称图G是强连通图</a></li>
<li><a href="#%E5%BC%BA%E8%BF%9E%E9%80%9A%E5%88%86%E9%87%8F%E6%9C%89%E5%90%91%E5%9B%BEg%E4%B8%AD%E7%9A%84%E6%9E%81%E5%A4%A7%E5%BC%BA%E8%BF%9E%E9%80%9A%E5%AD%90%E5%9B%BE%E7%A7%B0%E4%B8%BAg%E7%9A%84%E5%BC%BA%E8%BF%9E%E9%80%9A%E5%88%86%E9%87%8F%E6%98%BE%E7%84%B6%E5%BC%BA%E8%BF%9E%E9%80%9A%E5%9B%BE%E5%8F%AA%E6%9C%89%E4%B8%80%E4%B8%AA%E5%BC%BA%E8%BF%9E%E9%80%9A%E5%88%86%E9%87%8F%E5%8D%B3%E6%9C%AC%E8%BA%AB%E9%9D%9E%E5%BC%BA%E8%BF%9E%E9%80%9A%E5%9B%BE%E6%9C%89%E5%A4%9A%E4%B8%AA%E5%BC%BA%E8%BF%9E%E9%80%9A%E5%88%86%E9%87%8F">强连通分量：有向图G中的极大强连通子图称为G的强连通分量。显然，强连通图只有一个强连通分量，即本身。非强连通图有多个强连通分量</a></li>
<li><a href="#%E6%89%BE%E5%BC%BA%E8%BF%9E%E9%80%9A%E5%88%86%E9%87%8F%E7%9A%84%E6%96%B9%E6%B3%95">找强连通分量的方法</a></li>
</ul>
</li>
<li><a href="#%E5%9B%BE%E7%9A%84%E5%AD%98%E5%82%A8">图的存储</a><ul>
<li><a href="#%E9%82%BB%E6%8E%A5%E7%9F%A9%E9%98%B5%E5%94%AF%E4%B8%80">邻接矩阵（唯一）</a><ul>
<li><a href="#%E7%89%B9%E7%82%B9-2">特点</a></li>
<li><a href="#%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0-10">代码实现</a></li>
</ul>
</li>
<li><a href="#%E9%82%BB%E6%8E%A5%E8%A1%A8">邻接表</a><ul>
<li><a href="#%E5%AD%98%E5%82%A8%E6%96%B9%E5%BC%8F">存储方式</a></li>
<li><a href="#%E7%89%B9%E7%82%B9-3">特点</a></li>
<li><a href="#%E4%BB%A3%E7%A0%81%E5%AE%9E%E7%8E%B0-11">代码实现</a></li>
</ul>
</li>
<li><a href="#%E5%9B%BE%E7%9A%84%E5%9F%BA%E6%9C%AC%E8%BF%90%E7%AE%97">图的基本运算</a><ul>
<li><a href="#%E5%88%9B%E5%BB%BA%E9%82%BB%E6%8E%A5%E7%9F%A9%E9%98%B5-%E9%82%BB%E6%8E%A5%E8%A1%A8">创建(邻接矩阵-&gt;邻接表)</a></li>
<li><a href="#%E5%88%9B%E5%BB%BA%E9%82%BB%E6%8E%A5%E8%A1%A8-%E9%82%BB%E6%8E%A5%E7%9F%A9%E9%98%B5">创建(邻接表-&gt;邻接矩阵)</a></li>
</ul>
</li>
<li><a href="#%E5%B1%95%E7%A4%BA%E5%9B%BE%E7%9A%84%E9%82%BB%E6%8E%A5%E8%A1%A8">展示图的邻接表</a></li>
<li><a href="#%E5%9B%BE%E7%9A%84%E9%94%80%E6%AF%81">图的销毁</a></li>
<li><a href="#%E5%85%B6%E4%BB%96%E5%82%A8%E5%AD%98%E6%96%B9%E6%B3%95">其他储存方法</a><ul>
<li><a href="#%E5%8D%81%E5%AD%97%E9%93%BE%E8%A1%A8">十字链表</a></li>
<li><a href="#%E9%82%BB%E6%8E%A5%E5%A4%9A%E9%87%8D%E8%A1%A8">邻接多重表</a></li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- /TOC -->
<h1 id="树"><a href="#树" class="headerlink" title="树"></a>树</h1><h2 id="结点数-结点的度之和-分支数-1"><a href="#结点数-结点的度之和-分支数-1" class="headerlink" title="结点数 = 结点的度之和(分支数) + 1"></a>结点数 = 结点的度之和(分支数) + 1</h2><blockquote>
<p>【例（补充）】一棵度为4的树T中，若有20个度为4的结点，10个度为3的结点，1个度为2的结点，10个度为1的结点，则树T的叶子结点个数是()</p>
<blockquote>
<p>82</p>
</blockquote>
</blockquote>
<h2 id="度为m的树中第i层上至多有m-i-1-个结点-i≥1"><a href="#度为m的树中第i层上至多有m-i-1-个结点-i≥1" class="headerlink" title="度为m的树中第i层上至多有m^(i-1)个结点(i≥1)"></a>度为m的树中第i层上至多有m^(i-1)个结点(i≥1)</h2><h2 id="高度为h的m次树至多有-m-h-1-m-1-个结点-等比数列求和"><a href="#高度为h的m次树至多有-m-h-1-m-1-个结点-等比数列求和" class="headerlink" title="高度为h的m次树至多有(m^h-1)/(m-1)个结点(等比数列求和)"></a>高度为h的m次树至多有(m^h-1)/(m-1)个结点(等比数列求和)</h2><h2 id="具有n个结点的m次树的最小高度为logm-n-m-1-1-向上取整"><a href="#具有n个结点的m次树的最小高度为logm-n-m-1-1-向上取整" class="headerlink" title="具有n个结点的m次树的最小高度为logm(n(m-1)+1)向上取整"></a>具有n个结点的m次树的最小高度为logm(n(m-1)+1)向上取整</h2><hr>
<h2 id="树的存储结构"><a href="#树的存储结构" class="headerlink" title="树的存储结构"></a>树的存储结构</h2><h3 id="双亲存储结构"><a href="#双亲存储结构" class="headerlink" title="双亲存储结构"></a>双亲存储结构</h3><p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-10_10-58-01.png" alt="双亲存储结构"></p>
<h4 id="代码实现"><a href="#代码实现" class="headerlink" title="代码实现"></a>代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">typedef</span> <span class="class"><span class="keyword">struct</span> </span></span><br><span class="line"><span class="class">&#123;</span>  ElemType data;<span class="comment">//结点的值</span></span><br><span class="line">   <span class="keyword">int</span> parent;<span class="comment">//指向双亲的位置</span></span><br><span class="line">&#125; PTree[MaxSize];</span><br></pre></td></tr></table></figure>
<hr>
<h3 id="孩子链存储结构"><a href="#孩子链存储结构" class="headerlink" title="孩子链存储结构"></a>孩子链存储结构</h3><p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-10_10-58-012.png" alt="双亲存储结构"></p>
<h4 id="代码实现-1"><a href="#代码实现-1" class="headerlink" title="代码实现"></a>代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">typedef</span> <span class="class"><span class="keyword">struct</span> <span class="title">node</span></span></span><br><span class="line"><span class="class">&#123;</span></span><br><span class="line">   ElemType data;		      	<span class="comment">//结点的值</span></span><br><span class="line">   <span class="class"><span class="keyword">struct</span> <span class="title">node</span> *<span class="title">sons</span>[<span class="title">MaxSons</span>];</span>	<span class="comment">//指向孩子结点</span></span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<h3 id="求树高"><a href="#求树高" class="headerlink" title="求树高"></a>求树高</h3><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">TreeHeight</span><span class="params">(TSonNode *t)</span></span></span><br><span class="line"><span class="function"></span>&#123;  </span><br><span class="line">   TSonNode *p;</span><br><span class="line">   <span class="keyword">int</span> i,h,maxh=<span class="number">0</span>;</span><br><span class="line">   <span class="keyword">if</span> (t==<span class="literal">NULL</span>) <span class="keyword">return</span> <span class="number">0</span>;		<span class="comment">//空树返回高度0</span></span><br><span class="line">   <span class="keyword">else</span>					<span class="comment">//处理非空树</span></span><br><span class="line">   &#123;  <span class="keyword">for</span>（i=<span class="number">0</span>;i&lt;MaxSons;i++)</span><br><span class="line">      &#123;  p=t-&gt;sons[i];		<span class="comment">//p指向t的第i+1个孩子结点</span></span><br><span class="line">         <span class="keyword">if</span> (p!=<span class="literal">NULL</span>)			<span class="comment">//若存在第i+1个孩子</span></span><br><span class="line">	  &#123;  h=<span class="built_in">TreeHeight1</span>(p);	<span class="comment">//求出对应子树的高度</span></span><br><span class="line">            <span class="keyword">if</span> (maxh&lt;h) maxh=h;	<span class="comment">//求所有子树的最大高度</span></span><br><span class="line">	  &#125;</span><br><span class="line">      &#125;</span><br><span class="line">      <span class="keyword">return</span>(maxh+<span class="number">1</span>);			<span class="comment">//返回maxh+1</span></span><br><span class="line">   &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<hr>
<h2 id="孩子兄弟链存储结构"><a href="#孩子兄弟链存储结构" class="headerlink" title="孩子兄弟链存储结构"></a>孩子兄弟链存储结构</h2><p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-10_11-49-37.png" alt="孩子兄弟链存储结构"></p>
<h3 id="代码实现-2"><a href="#代码实现-2" class="headerlink" title="代码实现"></a>代码实现</h3><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">typedef</span> <span class="class"><span class="keyword">struct</span> <span class="title">tnode</span> </span></span><br><span class="line"><span class="class">&#123;</span>  ElemType data;		<span class="comment">//结点的值</span></span><br><span class="line">   <span class="class"><span class="keyword">struct</span> <span class="title">tnode</span> *<span class="title">hp</span>;</span>  	<span class="comment">//指向兄弟</span></span><br><span class="line">   <span class="class"><span class="keyword">struct</span> <span class="title">tnode</span> *<span class="title">vp</span>;</span>  	<span class="comment">//指向孩子结点</span></span><br><span class="line">&#125; TSBNode;</span><br><span class="line"></span><br></pre></td></tr></table></figure>
<h3 id="求树高度"><a href="#求树高度" class="headerlink" title="求树高度"></a>求树高度</h3><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">TreeHeight2</span><span class="params">(TSBNode *t)</span></span></span><br><span class="line"><span class="function"></span>&#123;  TSBNode *p;</span><br><span class="line">   <span class="keyword">int</span> h,maxh=<span class="number">0</span>;</span><br><span class="line">   <span class="keyword">if</span> (t==<span class="literal">NULL</span>) <span class="keyword">return</span> <span class="number">0</span>;		<span class="comment">//空树返回0</span></span><br><span class="line">   <span class="keyword">else</span></span><br><span class="line">   &#123;  p=t-&gt;vp;				<span class="comment">//指向第1个孩子结点</span></span><br><span class="line">      <span class="keyword">while</span> (p!=<span class="literal">NULL</span>)			<span class="comment">//扫描t的所有子树</span></span><br><span class="line">      &#123;  h=<span class="built_in">TreeHeight2</span>(p);		<span class="comment">//求出p子树的高度</span></span><br><span class="line">         <span class="keyword">if</span> (maxh&lt;h) maxh=h;		<span class="comment">//求所有子树的最大高度</span></span><br><span class="line">         p=p-&gt;hp;			<span class="comment">//继续处理t的其他子树</span></span><br><span class="line">      &#125;</span><br><span class="line">      <span class="keyword">return</span>(maxh+<span class="number">1</span>);			<span class="comment">//返回maxh+1</span></span><br><span class="line">   &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<hr>
<h1 id="二叉树"><a href="#二叉树" class="headerlink" title="二叉树"></a>二叉树</h1><h2 id="定义"><a href="#定义" class="headerlink" title="定义"></a>定义</h2><ol>
<li>是空（可以为空或者只含根节点）  </li>
<li>或者由一个根结点和两棵互不相交的称为左子树和右子树的二叉树组成。</li>
</ol>
<hr>
<h2 id="满二叉树"><a href="#满二叉树" class="headerlink" title="满二叉树"></a>满二叉树</h2><h2 id="高度为h的二叉树恰好有2-h-1-个结点"><a href="#高度为h的二叉树恰好有2-h-1-个结点" class="headerlink" title="高度为h的二叉树恰好有2^h-1 个结点"></a>高度为h的二叉树恰好有2^h-1 个结点</h2><h2 id="完全二叉树"><a href="#完全二叉树" class="headerlink" title="完全二叉树"></a>完全二叉树</h2><ol>
<li>最多只有下面两层的结点的度数小于2  </li>
<li>并且最下面一层的叶结点都依次排列在该层<strong>最左边</strong>的位置上<br>完全二叉树实际上是对应的满二叉树删除叶结点层<strong>最右边</strong>若干个结点得到的</li>
</ol>
<hr>
<h2 id="二叉树性质"><a href="#二叉树性质" class="headerlink" title="二叉树性质"></a>二叉树性质</h2><h3 id="普通二叉树性质"><a href="#普通二叉树性质" class="headerlink" title="普通二叉树性质"></a>普通二叉树性质</h3><ol>
<li>n0=n2+1（结点数 = 结点的度之和(分支数) + 1）<blockquote>
<p>一棵二叉树中有7个度为2的结点和5个度为1的结点，其总共有（   ）个结点</p>
<blockquote>
<p>20   </p>
</blockquote>
</blockquote>
</li>
</ol>
<blockquote>
<p>已知一棵完全二叉树的第6层（设根为第1层）有8个叶子结点，则该完全二叉树的结点个数最多是      </p>
<blockquote>
<p>111</p>
<blockquote>
<p>完全二叉树的叶子结点只能在最下两层，对于本题，结点最多的情况是第6层为倒数第二层，即1～6层构成一个满二叉树，其结点总数为26-1=63。其中第6层有25=32个结点，含8个叶子结点，则另外有32-8=24个非叶子结点，它们中每个结点有两个孩子结点（均为第7层的叶子结点），计48个叶子结点。这样最多的结点个数=63+48=111。</p>
</blockquote>
</blockquote>
</blockquote>
<h3 id="完全二叉树性质"><a href="#完全二叉树性质" class="headerlink" title="完全二叉树性质"></a>完全二叉树性质</h3><ol>
<li>n1可由n的奇偶性确定<br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84%5CSnipaste_2021-04-10_21-15-59.png"></li>
<li>若i ≤ |_ i/2 _|，则编号为i的结点为分支结点，否则为叶结点</li>
<li>双亲结点的编号为|_ i/2 _|</li>
<li><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84%5CSnipaste_2021-04-10_21-23-04.png"></li>
</ol>
<hr>
<h2 id="二叉树与树、森林之间的转换"><a href="#二叉树与树、森林之间的转换" class="headerlink" title="二叉树与树、森林之间的转换"></a>二叉树与树、森林之间的转换</h2><h3 id="森林、树转换为二叉树"><a href="#森林、树转换为二叉树" class="headerlink" title="森林、树转换为二叉树"></a>森林、树转换为二叉树</h3><ol>
<li>一颗树转换为二叉树<br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-10_21-27-26.png"></li>
<li>多颗树转换为一颗二叉树<br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-10_21-30-57.png"></li>
</ol>
<h3 id="二叉树还原为森林、树"><a href="#二叉树还原为森林、树" class="headerlink" title="二叉树还原为森林、树"></a>二叉树还原为森林、树</h3><ol>
<li>将一棵二叉树还原为一棵树<br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-10_21-35-31.png"></li>
<li>将一棵二叉树还原为多棵树<br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-10_21-37-41.png"><br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-10_21-38-14.png"></li>
</ol>
<hr>
<h2 id="二叉树的关系"><a href="#二叉树的关系" class="headerlink" title="二叉树的关系"></a>二叉树的关系</h2><p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-10_21-41-38.png"></p>
<h2 id="二叉树的存储结构"><a href="#二叉树的存储结构" class="headerlink" title="二叉树的存储结构"></a>二叉树的存储结构</h2><hr>
<h3 id="顺序存储"><a href="#顺序存储" class="headerlink" title="顺序存储"></a>顺序存储</h3><h4 id="图示"><a href="#图示" class="headerlink" title="图示"></a>图示</h4><p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-11_16-02-59.png"></p>
<h4 id="代码实现-3"><a href="#代码实现-3" class="headerlink" title="代码实现"></a>代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">typedef</span> ElemType SqBTree[MaxSize];</span><br><span class="line">SqBTree bt=<span class="string">&quot;#ABD#C#E######F&quot;</span>;</span><br></pre></td></tr></table></figure>
<h4 id="特点"><a href="#特点" class="headerlink" title="特点"></a>特点</h4><ol>
<li>对于<strong>完全二叉树</strong>而言十分合适，对于<strong>一般二叉树</strong>而言存在浪费</li>
<li>通过完全二叉树的性质可以找到双亲和孩子</li>
</ol>
<hr>
<h3 id="链式存储-二叉链"><a href="#链式存储-二叉链" class="headerlink" title="链式存储(二叉链)"></a>链式存储(二叉链)</h3><h4 id="图示-1"><a href="#图示-1" class="headerlink" title="图示"></a>图示</h4><p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-11_16-12-18.png"></p>
<h4 id="2-7-2-2-代码实现"><a href="#2-7-2-2-代码实现" class="headerlink" title="2.7.2.2. 代码实现"></a>2.7.2.2. 代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">typedef</span> <span class="class"><span class="keyword">struct</span> <span class="title">node</span></span></span><br><span class="line"><span class="class">&#123;</span></span><br><span class="line">   ElemType data;</span><br><span class="line">   node *left_child,*right_child;</span><br><span class="line">&#125;BTNode;</span><br></pre></td></tr></table></figure>
<h4 id="2-7-2-3-特点"><a href="#2-7-2-3-特点" class="headerlink" title="2.7.2.3. 特点"></a>2.7.2.3. 特点</h4><ol>
<li>节约空间</li>
<li>找孩子容易</li>
<li>找双亲不容易</li>
</ol>
<hr>
<h2 id="2-8-二叉树基本运算"><a href="#2-8-二叉树基本运算" class="headerlink" title="2.8. 二叉树基本运算"></a>2.8. 二叉树基本运算</h2><h3 id="2-8-1-创建二叉树"><a href="#2-8-1-创建二叉树" class="headerlink" title="2.8.1. 创建二叉树"></a>2.8.1. 创建二叉树</h3><h4 id="2-8-1-1-算法设计"><a href="#2-8-1-1-算法设计" class="headerlink" title="2.8.1.1. 算法设计"></a>2.8.1.1. 算法设计</h4><ol>
<li>若ch=‘(’：则将前面刚创建的结点作为双亲结点进栈，并置k=1，表示开始处理左孩子结点；</li>
<li>若ch=‘)’：表示栈顶结点的左、右孩子结点处理完毕，退栈；</li>
<li>若ch=‘，’：表示开始处理右孩子结点，置k=2；</li>
<li>其他情况（结点值）：<ol>
<li>创建*p结点用于存放ch；  </li>
<li>当k=1时，将*p结点作为栈顶结点的左孩子结点；</li>
<li>当k=2时，将*p结点作为栈顶结点的右孩子结点。</li>
</ol>
</li>
</ol>
<h4 id="2-8-1-2-代码实现"><a href="#2-8-1-2-代码实现" class="headerlink" title="2.8.1.2. 代码实现"></a>2.8.1.2. 代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">CreateBTree</span><span class="params">(BTNode * &amp;b,<span class="keyword">char</span> *str)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    BTNode *st[MaxSize],*p;</span><br><span class="line">    <span class="keyword">int</span> top = <span class="number">-1</span>,k,j = <span class="number">0</span>;</span><br><span class="line">    <span class="keyword">char</span> ch;</span><br><span class="line">    ch = str[<span class="number">0</span>];</span><br><span class="line">    b = <span class="literal">NULL</span>;</span><br><span class="line">    <span class="keyword">while</span>(ch!=<span class="string">&#x27;\0&#x27;</span>)</span><br><span class="line">    &#123;</span><br><span class="line">        <span class="built_in"><span class="keyword">switch</span></span> (ch)</span><br><span class="line">        &#123;</span><br><span class="line">            <span class="keyword">case</span> <span class="string">&#x27;(&#x27;</span>:</span><br><span class="line">                top++;</span><br><span class="line">                st[top] = p;</span><br><span class="line">                k = <span class="number">1</span>;</span><br><span class="line">                <span class="keyword">break</span>;</span><br><span class="line"></span><br><span class="line">            <span class="keyword">case</span> <span class="string">&#x27;)&#x27;</span>:</span><br><span class="line">                top--;</span><br><span class="line">                <span class="keyword">break</span>;</span><br><span class="line"></span><br><span class="line">            <span class="keyword">case</span> <span class="string">&#x27;,&#x27;</span>:</span><br><span class="line">                k = <span class="number">2</span>;</span><br><span class="line">                <span class="keyword">break</span>;</span><br><span class="line"></span><br><span class="line">            <span class="keyword">default</span>:</span><br><span class="line">                p = (BTNode *)<span class="built_in">malloc</span>(<span class="built_in"><span class="keyword">sizeof</span></span>(BTNode));</span><br><span class="line">                p-&gt;data = ch ;</span><br><span class="line">                p-&gt;left_child = p-&gt;right_child = <span class="literal">NULL</span>;</span><br><span class="line">                <span class="keyword">if</span>(b == <span class="literal">NULL</span>)<span class="comment">//此时p为二叉树的根结点</span></span><br><span class="line">                   b=p;</span><br><span class="line">                <span class="keyword">else</span></span><br><span class="line">                &#123;</span><br><span class="line">                    <span class="built_in"><span class="keyword">switch</span></span> (k)</span><br><span class="line">                    &#123;</span><br><span class="line">                    <span class="keyword">case</span> <span class="number">1</span>:</span><br><span class="line">                        st[top]-&gt;left_child = p;</span><br><span class="line">                        <span class="keyword">break</span>;</span><br><span class="line">                    <span class="keyword">case</span> <span class="number">2</span>:</span><br><span class="line">                        st[top]-&gt;right_child = p;</span><br><span class="line">                        <span class="keyword">break</span>;</span><br><span class="line">                    <span class="keyword">default</span>:</span><br><span class="line">                       <span class="keyword">break</span>;</span><br><span class="line">                &#125;</span><br><span class="line">            &#125;</span><br><span class="line">            <span class="keyword">break</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        j++;</span><br><span class="line">        ch = str[j];</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br><span class="line"></span><br></pre></td></tr></table></figure>
<hr>
<h3 id="2-8-2-销毁二叉树"><a href="#2-8-2-销毁二叉树" class="headerlink" title="2.8.2. 销毁二叉树"></a>2.8.2. 销毁二叉树</h3><h4 id="2-8-2-1-算法设计"><a href="#2-8-2-1-算法设计" class="headerlink" title="2.8.2.1. 算法设计"></a>2.8.2.1. 算法设计</h4><p>递归模型：</p>
<ol>
<li>if(b==NULL) 返回;</li>
<li>else<br>f(b-&gt;left_child);<br>f(b-&gt;right_child);<br>释放*b结点;</li>
</ol>
<h4 id="2-8-2-2-代码实现"><a href="#2-8-2-2-代码实现" class="headerlink" title="2.8.2.2. 代码实现"></a>2.8.2.2. 代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">DestroyBTree</span><span class="params">(BTNode *&amp;b)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">if</span>(b == <span class="literal">NULL</span>) </span><br><span class="line">        <span class="keyword">return</span> ;</span><br><span class="line">    <span class="keyword">else</span></span><br><span class="line">    &#123;</span><br><span class="line">        <span class="built_in">DestroyBTree</span>(b-&gt;left_child);</span><br><span class="line">        <span class="built_in">DestroyBTree</span>(b-&gt;right_child);</span><br><span class="line">        <span class="built_in">free</span>(b);</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br><span class="line"></span><br></pre></td></tr></table></figure>
<hr>
<h3 id="2-8-3-查找结点"><a href="#2-8-3-查找结点" class="headerlink" title="2.8.3. 查找结点"></a>2.8.3. 查找结点</h3><h4 id="2-8-3-1-算法设计"><a href="#2-8-3-1-算法设计" class="headerlink" title="2.8.3.1. 算法设计"></a>2.8.3.1. 算法设计</h4><p>递归模型：</p>
<ol>
<li>if(b==NULL)<br>f(b，x) = NULL;</li>
<li>if(b-&gt;data==x)<br>f(b，x) = b  </li>
<li>if(p=f(b-&gt;lchild，x)且p!=NULL)//若在左子树中找到了<br>f(b，x) = p</li>
<li>else<br>f(b，x) = f(b-&gt;rchild，x)</li>
</ol>
<h4 id="2-8-3-2-代码实现"><a href="#2-8-3-2-代码实现" class="headerlink" title="2.8.3.2. 代码实现"></a>2.8.3.2. 代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br></pre></td><td class="code"><pre><span class="line"><span class="function">BTNode *<span class="title">FindBTNode</span><span class="params">(BTNode *b,<span class="keyword">char</span> a)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    BTNode *p;</span><br><span class="line">    <span class="keyword">if</span>(b == <span class="literal">NULL</span>) </span><br><span class="line">        <span class="keyword">return</span> <span class="literal">NULL</span>;</span><br><span class="line">    <span class="keyword">else</span> <span class="keyword">if</span>(b-&gt;data == a)</span><br><span class="line">        <span class="keyword">return</span> b;</span><br><span class="line">    <span class="keyword">else</span></span><br><span class="line">    &#123;</span><br><span class="line">            p = <span class="built_in">FindBTNode</span>(b-&gt;left_child,a);</span><br><span class="line">            <span class="keyword">if</span>(p!=<span class="literal">NULL</span>)</span><br><span class="line">                <span class="keyword">return</span> p;</span><br><span class="line">            <span class="keyword">else</span></span><br><span class="line">            &#123;</span><br><span class="line">                <span class="keyword">return</span> <span class="built_in">FindBTNode</span>(b-&gt;right_child,a);</span><br><span class="line">            &#125;</span><br><span class="line">            </span><br><span class="line">    &#125; </span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<hr>
<h3 id="2-8-4-求高度"><a href="#2-8-4-求高度" class="headerlink" title="2.8.4. 求高度"></a>2.8.4. 求高度</h3><h4 id="2-8-4-1-代码实现"><a href="#2-8-4-1-代码实现" class="headerlink" title="2.8.4.1. 代码实现"></a>2.8.4.1. 代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">BTDepth</span><span class="params">(BTNode *b)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">int</span> left_child_depth,right_child_depth;</span><br><span class="line">    <span class="keyword">if</span>(b == <span class="literal">NULL</span>)</span><br><span class="line">        <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">    <span class="keyword">else</span></span><br><span class="line">    &#123;</span><br><span class="line">        left_child_depth = <span class="built_in">BTDepth</span>(b-&gt;left_child);</span><br><span class="line">        right_child_depth = <span class="built_in">BTDepth</span>(b-&gt;right_child);</span><br><span class="line">        <span class="keyword">return</span> (left_child_depth&gt;right_child_depth)?(left_child_depth+<span class="number">1</span>):(right_child_depth+<span class="number">1</span>);</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<hr>
<h3 id="2-8-5-输出二叉树"><a href="#2-8-5-输出二叉树" class="headerlink" title="2.8.5. 输出二叉树"></a>2.8.5. 输出二叉树</h3><h4 id="2-8-5-1-代码实现"><a href="#2-8-5-1-代码实现" class="headerlink" title="2.8.5.1. 代码实现"></a>2.8.5.1. 代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">DisplayBTree</span><span class="params">(BTNode *b)</span></span></span><br><span class="line"><span class="function"></span>&#123;  <span class="keyword">if</span> (b!=<span class="literal">NULL</span>)</span><br><span class="line">   &#123;  <span class="built_in">printf</span>(<span class="string">&quot;%c&quot;</span>,b-&gt;data);</span><br><span class="line">      <span class="keyword">if</span> (b-&gt;left_child!=<span class="literal">NULL</span> || b-&gt;right_child!=<span class="literal">NULL</span>)</span><br><span class="line">      &#123;  <span class="built_in">printf</span>(<span class="string">&quot;(&quot;</span>);</span><br><span class="line">	  <span class="built_in">DisplayBTree</span>(b-&gt;left_child);<span class="comment">//递归处理左子树</span></span><br><span class="line">	  <span class="keyword">if</span> (b-&gt;right_child!=<span class="literal">NULL</span>) <span class="built_in">printf</span>(<span class="string">&quot;，&quot;</span>);</span><br><span class="line">	  <span class="built_in">DisplayBTree</span>(b-&gt;right_child);<span class="comment">//递归处理右子树</span></span><br><span class="line">	  <span class="built_in">printf</span>(<span class="string">&quot;)&quot;</span>);</span><br><span class="line">      &#125;</span><br><span class="line">   &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<hr>
<h2 id="2-9-二叉树的遍历"><a href="#2-9-二叉树的遍历" class="headerlink" title="2.9. 二叉树的遍历"></a>2.9. 二叉树的遍历</h2><h3 id="2-9-1-先序遍历"><a href="#2-9-1-先序遍历" class="headerlink" title="2.9.1. 先序遍历"></a>2.9.1. 先序遍历</h3><blockquote>
<p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-14_09-47-58.png"></p>
<blockquote>
<p>顺序：A B D G C E F</p>
</blockquote>
</blockquote>
<h3 id="2-9-2-代码实现"><a href="#2-9-2-代码实现" class="headerlink" title="2.9.2. 代码实现"></a>2.9.2. 代码实现</h3><ol>
<li>递归算法<figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">PreOrder</span><span class="params">(BTNode *b)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">if</span>(b != <span class="literal">NULL</span>)</span><br><span class="line">    &#123;</span><br><span class="line">        cout&lt;&lt;b-&gt;data&lt;&lt;endl;</span><br><span class="line">        <span class="built_in">PreOrder</span>(b-&gt;left_child);</span><br><span class="line">        <span class="built_in">PreOrder</span>(b-&gt;right_child);</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br><span class="line"></span><br></pre></td></tr></table></figure></li>
<li>非归算法<figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">PreOrder1</span><span class="params">(BTNode *b)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    BTNode *p;</span><br><span class="line">    SqStack *st;</span><br><span class="line">    <span class="built_in">InitStack</span>(st); </span><br><span class="line">    <span class="keyword">if</span>(b!=<span class="literal">NULL</span>)</span><br><span class="line">    &#123;</span><br><span class="line">        <span class="built_in">Push</span>(st,b);</span><br><span class="line">        <span class="keyword">while</span>(!<span class="built_in">StackEmpty</span>(st))</span><br><span class="line">        &#123;</span><br><span class="line">            <span class="built_in">Pop</span>(st,p);</span><br><span class="line">            cout&lt;&lt;p-&gt;data&lt;&lt;endl;</span><br><span class="line">            <span class="keyword">if</span>(p-&gt;right_child! = <span class="literal">NULL</span>)</span><br><span class="line">                <span class="built_in">Push</span>(st,p-&gt;right_child);</span><br><span class="line">            <span class="keyword">if</span>(p-&gt;right_child! = <span class="literal">NULL</span>)</span><br><span class="line">                <span class="built_in">Push</span>(st,p-&gt;left_child);</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;   </span><br><span class="line">    <span class="built_in">DestroyStack</span>(st);</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure></li>
</ol>
<hr>
<h3 id="2-9-3-中序遍历"><a href="#2-9-3-中序遍历" class="headerlink" title="2.9.3. 中序遍历"></a>2.9.3. 中序遍历</h3><blockquote>
<p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-14_09-47-58.png"></p>
<blockquote>
<p>顺序：D G B A E C F</p>
</blockquote>
</blockquote>
<h3 id="2-9-4-代码实现"><a href="#2-9-4-代码实现" class="headerlink" title="2.9.4. 代码实现"></a>2.9.4. 代码实现</h3><ol>
<li>递归算法<figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">InOrder</span><span class="params">(BTNode *b)</span></span></span><br><span class="line"><span class="function"></span>&#123;  <span class="keyword">if</span> (b!=<span class="literal">NULL</span>)  </span><br><span class="line">   &#123;  </span><br><span class="line">        <span class="built_in">InOrder</span>(b-&gt;left_child);</span><br><span class="line">        cout&lt;&lt;b-&gt;data&lt;&lt;endl;</span><br><span class="line">        <span class="built_in">InOrder</span>(b-&gt;right_child);</span><br><span class="line">   &#125;</span><br><span class="line">&#125;</span><br><span class="line"></span><br></pre></td></tr></table></figure></li>
<li>非归算法<figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">InOrder1</span><span class="params">(BTNode *b)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    BTNode *p;</span><br><span class="line">    SqStack *st;</span><br><span class="line">    p =b;</span><br><span class="line">    <span class="keyword">while</span>(<span class="built_in">StackEmpty</span>(st)|| p != <span class="literal">NULL</span>)</span><br><span class="line">    &#123;</span><br><span class="line">        <span class="keyword">while</span>(p != <span class="literal">NULL</span>)</span><br><span class="line">        &#123;</span><br><span class="line">            <span class="built_in">Push</span>(st,p);</span><br><span class="line">            p = p-&gt;left_child;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">if</span>(!<span class="built_in">StackEmpty</span>(st))</span><br><span class="line">        &#123;</span><br><span class="line">            <span class="built_in">Pop</span>(st,p);</span><br><span class="line">            cout&lt;&lt;p-&gt;data&lt;&lt;endl;</span><br><span class="line">            p = p-&gt;right_child;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="built_in">DestroyStack</span>(st);</span><br><span class="line">&#125;</span><br><span class="line"></span><br></pre></td></tr></table></figure></li>
</ol>
<hr>
<h3 id="2-9-5-后序遍历"><a href="#2-9-5-后序遍历" class="headerlink" title="2.9.5. 后序遍历"></a>2.9.5. 后序遍历</h3><blockquote>
<p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-14_09-47-58.png"></p>
<blockquote>
<p>顺序：G D B E F C A</p>
</blockquote>
</blockquote>
<h3 id="2-9-6-代码实现"><a href="#2-9-6-代码实现" class="headerlink" title="2.9.6. 代码实现"></a>2.9.6. 代码实现</h3><ol>
<li>递归算法<figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">PostOrder</span><span class="params">(BTNode *b)</span> </span></span><br><span class="line"><span class="function"></span>&#123;  <span class="keyword">if</span> (b!=<span class="literal">NULL</span>)  </span><br><span class="line">   &#123;  </span><br><span class="line">        <span class="built_in">PostOrder</span>(b-&gt;left_child);</span><br><span class="line">        <span class="built_in">PostOrder</span>(b-&gt;right_child);</span><br><span class="line">        cout&lt;&lt;b-&gt;data&lt;&lt;endl;</span><br><span class="line">   &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure></li>
<li>非归算法<figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">PostOrder1</span><span class="params">(BTNode *b)</span>	<span class="comment">//后序非递归遍历算法</span></span></span><br><span class="line"><span class="function"></span>&#123;  BTNode *p，*r;</span><br><span class="line">   <span class="keyword">bool</span> flag;</span><br><span class="line">   SqStack *st;		<span class="comment">//定义一个顺序栈指针st</span></span><br><span class="line">   <span class="built_in">InitStack</span>(st);		<span class="comment">//初始化栈st</span></span><br><span class="line">   p=b;</span><br><span class="line">   <span class="keyword">do</span></span><br><span class="line">   &#123;  <span class="keyword">while</span> (p!=<span class="literal">NULL</span>)		<span class="comment">//扫描结点p的所有左下结点并进栈</span></span><br><span class="line">      &#123;  <span class="built_in">Push</span>(st，p);		<span class="comment">//结点p进栈</span></span><br><span class="line">	  p=p-&gt;lchild;		<span class="comment">//移动到左孩子</span></span><br><span class="line">      &#125;</span><br><span class="line">      r=<span class="literal">NULL</span>;			<span class="comment">//r指向刚刚访问的结点，初始时为空</span></span><br><span class="line">      flag=<span class="literal">true</span>;		<span class="comment">//flag为真表示正在处理栈顶结点</span></span><br><span class="line">     <span class="keyword">while</span> (!<span class="built_in">StackEmpty</span>(st) &amp;&amp; flag)</span><br><span class="line">     &#123;  <span class="built_in">GetTop</span>(st，p);	         <span class="comment">//取出当前的栈顶结点p</span></span><br><span class="line">        <span class="keyword">if</span> (p-&gt;rchild==r)      <span class="comment">//若结点p的右孩子为空或者为刚访问结点       </span></span><br><span class="line">        &#123;  <span class="built_in">printf</span>(<span class="string">&quot;%c &quot;</span>，p-&gt;data);     <span class="comment">//访问结点p</span></span><br><span class="line">	    <span class="built_in">Pop</span>(st，p);</span><br><span class="line">	    r=p;		 <span class="comment">//r指向刚访问过的结点</span></span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">else</span></span><br><span class="line">        &#123;  p=p-&gt;rchild;	<span class="comment">//转向处理其右子树</span></span><br><span class="line">	    flag=<span class="literal">false</span>;	<span class="comment">//表示当前不是处理栈顶结点</span></span><br><span class="line">        &#125;</span><br><span class="line">     &#125;</span><br><span class="line">   &#125; <span class="keyword">while</span> (!<span class="built_in">StackEmpty</span>(st));	<span class="comment">//栈不空循环</span></span><br><span class="line">   <span class="built_in">printf</span>(<span class="string">&quot;\n&quot;</span>);</span><br><span class="line">   <span class="built_in">DestroyStack</span>(st);		<span class="comment">//销毁栈</span></span><br><span class="line">&#125;</span><br><span class="line"></span><br></pre></td></tr></table></figure></li>
</ol>
<hr>
<h2 id="2-10-二叉树的构造"><a href="#2-10-二叉树的构造" class="headerlink" title="2.10. 二叉树的构造"></a>2.10. 二叉树的构造</h2><h3 id="2-10-1-条件"><a href="#2-10-1-条件" class="headerlink" title="2.10.1. 条件"></a>2.10.1. 条件</h3><p>中序+前序/后序，可唯一确定。</p>
<h3 id="2-10-2-中序-前序"><a href="#2-10-2-中序-前序" class="headerlink" title="2.10.2. 中序 + 前序"></a>2.10.2. 中序 + 前序</h3><h4 id="2-10-2-1-递归原理"><a href="#2-10-2-1-递归原理" class="headerlink" title="2.10.2.1. 递归原理"></a>2.10.2.1. 递归原理</h4><p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-14_20-14-14.png"><br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-14_20-12-51.png"></p>
<h4 id="2-10-2-2-代码实现"><a href="#2-10-2-2-代码实现" class="headerlink" title="2.10.2.2. 代码实现"></a>2.10.2.2. 代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br></pre></td><td class="code"><pre><span class="line"><span class="function">BTNode *<span class="title">Pre_MidCreate</span><span class="params">(<span class="keyword">char</span> *pre,<span class="keyword">char</span> *mid,<span class="keyword">int</span> n)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    BTNode *result;</span><br><span class="line">    <span class="keyword">char</span> *p;</span><br><span class="line">    <span class="keyword">int</span> k;</span><br><span class="line">    <span class="keyword">if</span>(n&lt;=<span class="number">0</span>) <span class="keyword">return</span> <span class="literal">NULL</span>;</span><br><span class="line">    result = (BTNode*)<span class="built_in">malloc</span>(<span class="built_in"><span class="keyword">sizeof</span></span>(BTNode));</span><br><span class="line">    result-&gt;data = *pre;</span><br><span class="line">    <span class="keyword">for</span>(p = mid; p &lt; mid + n;p++)</span><br><span class="line">        <span class="keyword">if</span>(*p == *pre)</span><br><span class="line">            <span class="keyword">break</span>;</span><br><span class="line">    k = p - mid;</span><br><span class="line">    result-&gt;left_child = <span class="built_in">Pre_MidCreate</span>(pre+<span class="number">1</span>,mid,k);</span><br><span class="line">    result-&gt;right_child = <span class="built_in">Pre_MidCreate</span>(pre+k+<span class="number">1</span>,p+<span class="number">1</span>,n-k<span class="number">-1</span>);</span><br><span class="line">    <span class="keyword">return</span> result;</span><br><span class="line">&#125;</span><br><span class="line"></span><br></pre></td></tr></table></figure>
<hr>
<h3 id="2-10-3-中序-后序"><a href="#2-10-3-中序-后序" class="headerlink" title="2.10.3. 中序 + 后序"></a>2.10.3. 中序 + 后序</h3><h4 id="2-10-3-1-递归原理"><a href="#2-10-3-1-递归原理" class="headerlink" title="2.10.3.1. 递归原理"></a>2.10.3.1. 递归原理</h4><p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-14_20-15-36.png"><br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-14_20-16-05.png"></p>
<h4 id="2-10-3-2-代码实现"><a href="#2-10-3-2-代码实现" class="headerlink" title="2.10.3.2. 代码实现"></a>2.10.3.2. 代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br></pre></td><td class="code"><pre><span class="line"><span class="function">BTNode *<span class="title">Post_MidCreate</span><span class="params">(<span class="keyword">char</span> *post,<span class="keyword">char</span> *mid,<span class="keyword">int</span> n)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    BTNode *result;</span><br><span class="line">    <span class="keyword">char</span> r,*p;</span><br><span class="line">    <span class="keyword">int</span> k;</span><br><span class="line">    <span class="keyword">if</span>(n&lt;=<span class="number">0</span>) <span class="keyword">return</span> <span class="literal">NULL</span>;</span><br><span class="line">    r = *(post+n<span class="number">-1</span>);</span><br><span class="line">    result = (BTNode*)<span class="built_in">malloc</span>(<span class="built_in"><span class="keyword">sizeof</span></span>(BTNode));</span><br><span class="line">    result-&gt;data = r;</span><br><span class="line">    <span class="keyword">for</span>(p = mid;p&lt;mid+n;p++)</span><br><span class="line">        <span class="keyword">if</span>(*p == r)</span><br><span class="line">            <span class="keyword">break</span>;</span><br><span class="line">    k = p - mid;</span><br><span class="line">    result-&gt;left_child = <span class="built_in">Post_MidCreate</span>(post,mid,k);</span><br><span class="line">    result-&gt;right_child = <span class="built_in">Post_MidCreate</span>(post+k,p+<span class="number">1</span>,n-k<span class="number">-1</span>);</span><br><span class="line">    <span class="keyword">return</span> result;</span><br><span class="line">&#125;</span><br><span class="line"></span><br></pre></td></tr></table></figure>
<hr>
<h2 id="线索二叉树"><a href="#线索二叉树" class="headerlink" title="线索二叉树"></a>线索二叉树</h2><h3 id="结构"><a href="#结构" class="headerlink" title="结构"></a>结构</h3><p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-23_19-34-16.png"><br>ltag和rtag是左右标志，0表示指向节点，1表示指向线索<br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-23_19-41-33.png"></p>
<h4 id="代码实现-4"><a href="#代码实现-4" class="headerlink" title="代码实现"></a>代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">typedef</span> <span class="class"><span class="keyword">struct</span> <span class="title">node</span></span></span><br><span class="line"><span class="class">&#123;</span></span><br><span class="line">    <span class="keyword">char</span> data;</span><br><span class="line">    <span class="keyword">int</span> right_tag,left_tag;</span><br><span class="line">    node *left_child;</span><br><span class="line">    node *right_child;</span><br><span class="line">&#125;TBTNode;</span><br><span class="line"></span><br></pre></td></tr></table></figure>
<h3 id="线索化二叉树"><a href="#线索化二叉树" class="headerlink" title="线索化二叉树"></a>线索化二叉树</h3><h4 id="方法"><a href="#方法" class="headerlink" title="方法"></a>方法</h4><ol>
<li>CreatThread(b)算法：对以二叉链存储的二叉树b进行中序线索化，并返回线索化后头结点的指针root。</li>
<li>Thread(p)算法：对以p为根结点的二叉树子树的中序线索化。   </li>
</ol>
<h4 id="代码实现-5"><a href="#代码实现-5" class="headerlink" title="代码实现"></a>代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br></pre></td><td class="code"><pre><span class="line">TBTNode *pre;<span class="comment">//全局变量</span></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">Thread</span><span class="params">(TBTNode *&amp;p)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">if</span>(p!=<span class="literal">NULL</span>)</span><br><span class="line">    &#123;</span><br><span class="line">        <span class="built_in">Thread</span>(p-&gt;left_child);</span><br><span class="line">        <span class="keyword">if</span>(p-&gt;left_child == <span class="literal">NULL</span>)<span class="comment">//建立当前结点的前驱线索</span></span><br><span class="line">        &#123;</span><br><span class="line">            p-&gt;left_child = pre;</span><br><span class="line">            p-&gt;left_tag = <span class="number">1</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">else</span></span><br><span class="line">        &#123;</span><br><span class="line">            p-&gt;left_tag = <span class="number">0</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">if</span>(pre-&gt;right_child == <span class="literal">NULL</span>)<span class="comment">//建立前驱结点的后继线索</span></span><br><span class="line">        &#123;</span><br><span class="line">            pre-&gt;right_child = p;</span><br><span class="line">            pre-&gt;right_tag = <span class="number">1</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">else</span></span><br><span class="line">        &#123;</span><br><span class="line">            pre-&gt;right_tag = <span class="number">0</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        pre = p;<span class="comment">//将全局变量pre后推一位</span></span><br><span class="line">        <span class="built_in">Thread</span>(p-&gt;right_child); </span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">&#125;</span><br><span class="line"><span class="function">TBTNode *<span class="title">CreateThread</span><span class="params">(TBTNode *b)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    TBTNode *root;</span><br><span class="line">    root = (TBTNode *)<span class="built_in">malloc</span>(<span class="built_in"><span class="keyword">sizeof</span></span>(TBTNode));</span><br><span class="line">    root-&gt;left_tag = <span class="number">0</span>;</span><br><span class="line">    root-&gt;right_tag = <span class="number">1</span>;</span><br><span class="line">    root-&gt;right_child = b;</span><br><span class="line">    <span class="keyword">if</span>(b == <span class="literal">NULL</span>)</span><br><span class="line">        root-&gt;left_child = root;</span><br><span class="line">    <span class="keyword">else</span></span><br><span class="line">    &#123;</span><br><span class="line">        root-&gt;left_child = b;</span><br><span class="line">        pre = root;</span><br><span class="line">        <span class="built_in">Thread</span>(b);</span><br><span class="line">        <span class="comment">//处理头节点root的指向</span></span><br><span class="line">        pre-&gt;right_child = root;</span><br><span class="line">        pre-&gt;right_tag = <span class="number">1</span>;</span><br><span class="line">        root-&gt;right_child = pre;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">return</span> root;</span><br><span class="line">&#125;</span><br><span class="line"></span><br></pre></td></tr></table></figure>

<hr>
<h2 id="哈夫曼树"><a href="#哈夫曼树" class="headerlink" title="哈夫曼树"></a>哈夫曼树</h2><h3 id="计算WPL"><a href="#计算WPL" class="headerlink" title="计算WPL"></a>计算WPL</h3><h4 id="根结点到各个叶结点的路径长度与相应结点权值的乘积的和，叫做二叉树的带权路径长度WPL。"><a href="#根结点到各个叶结点的路径长度与相应结点权值的乘积的和，叫做二叉树的带权路径长度WPL。" class="headerlink" title="根结点到各个叶结点的路径长度与相应结点权值的乘积的和，叫做二叉树的带权路径长度WPL。"></a>根结点到各个叶结点的路径长度与相应结点权值的乘积的和，叫做二叉树的带权路径长度WPL。</h4><blockquote>
<p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-23_20-07-07.png"></p>
<blockquote>
<p>WPL(T1) =<br>7 * 2+5 * 2+2 * 3+4 * 3+9 *2  =60</p>
</blockquote>
</blockquote>
<h3 id="构造哈夫曼树"><a href="#构造哈夫曼树" class="headerlink" title="构造哈夫曼树"></a>构造哈夫曼树</h3><h4 id><a href="#" class="headerlink" title></a></h4><blockquote>
<p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-23_20-09-42.png"></p>
<blockquote>
<p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-23_20-15-18.png"><br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-23_20-16-00.png"></p>
</blockquote>
</blockquote>
<h3 id="哈夫曼编码"><a href="#哈夫曼编码" class="headerlink" title="哈夫曼编码"></a>哈夫曼编码</h3><h4 id="特点-1"><a href="#特点-1" class="headerlink" title="特点"></a>特点</h4><p>权值越大的字符编码越短，反之越长<br><strong>不可能出现</strong>一个字符的哈夫曼编码是另一个字符哈夫曼编码的<strong>前缀</strong></p>
<h4 id="示例"><a href="#示例" class="headerlink" title="示例"></a>示例</h4><p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-23_20-12-19.png"></p>
<hr>
<h2 id="并查集"><a href="#并查集" class="headerlink" title="并查集"></a>并查集</h2><h3 id="构造"><a href="#构造" class="headerlink" title="构造"></a>构造</h3><blockquote>
<p>2 4/<br>5 7/<br>1 3/<br>8 9/<br>1 2/<br>5 6/<br>2 3</p>
<blockquote>
<p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-28_19-25-13.png"><br>4个集合{1，2，3，4}、{5，6，7}、{8，9}、{10}，分别以4、7、9和10表示对应集合的编号。<br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-28_19-29-59.png"></p>
</blockquote>
</blockquote>
<h3 id="数据结构"><a href="#数据结构" class="headerlink" title="数据结构"></a>数据结构</h3><h4 id="代码实现-6"><a href="#代码实现-6" class="headerlink" title="代码实现"></a>代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">typedef</span> <span class="class"><span class="keyword">struct</span> <span class="title">node</span></span></span><br><span class="line"><span class="class">&#123;</span></span><br><span class="line">    <span class="keyword">int</span> data;</span><br><span class="line">    <span class="keyword">int</span> rank;<span class="comment">//结点秩，大致为树的高度</span></span><br><span class="line">    <span class="keyword">int</span> parent;<span class="comment">//结点对应双亲下标</span></span><br><span class="line">&#125;UFSTree;</span><br></pre></td></tr></table></figure>
<h3 id="初始化"><a href="#初始化" class="headerlink" title="初始化"></a>初始化</h3><h4 id="代码实现-7"><a href="#代码实现-7" class="headerlink" title="代码实现"></a>代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">Make_Set</span><span class="params">(UFSTree t[],<span class="keyword">int</span> n)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">int</span> i;</span><br><span class="line">    <span class="keyword">for</span>(i = <span class="number">1</span>;i&lt;=n;i++)</span><br><span class="line">    &#123;</span><br><span class="line">        t[i].data = <span class="number">-1</span>;</span><br><span class="line">        t[i].rank = <span class="number">0</span>;<span class="comment">//树高为0</span></span><br><span class="line">        t[i].parent = i;<span class="comment">//指向自己</span></span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<h3 id="查找一个元素所属的集合"><a href="#查找一个元素所属的集合" class="headerlink" title="查找一个元素所属的集合"></a>查找一个元素所属的集合</h3><h4 id="代码实现-8"><a href="#代码实现-8" class="headerlink" title="代码实现"></a>代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">Find_Set</span><span class="params">(UFSTree t[],<span class="keyword">int</span> x)</span><span class="comment">//在x所在子树中查找集合编号</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">if</span>(x != t[x].parent)</span><br><span class="line">        <span class="keyword">return</span>(<span class="built_in">Find_Set</span>(t,t[x].parent));</span><br><span class="line">    <span class="keyword">else</span></span><br><span class="line">        <span class="keyword">return</span> x;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<h3 id="两个元素各自所属的集合的合并"><a href="#两个元素各自所属的集合的合并" class="headerlink" title="两个元素各自所属的集合的合并"></a>两个元素各自所属的集合的合并</h3><h4 id="代码实现-9"><a href="#代码实现-9" class="headerlink" title="代码实现"></a>代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">Union_Set</span><span class="params">(UFSTree t[],<span class="keyword">int</span> x,<span class="keyword">int</span> y)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    x = <span class="built_in">Find_Set</span>(t,x);</span><br><span class="line">    y = <span class="built_in">Find_Set</span>(t,y);</span><br><span class="line">    <span class="keyword">if</span>(t[x].rank &gt; t[y].rank)</span><br><span class="line">    t[y].parent = x;</span><br><span class="line">    <span class="keyword">else</span></span><br><span class="line">    &#123;</span><br><span class="line">        t[x].parent = y;</span><br><span class="line">        <span class="keyword">if</span>(t[x].parent == t[y].parent)</span><br><span class="line">            t[y].rank++;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<hr>
<h1 id="图"><a href="#图" class="headerlink" title="图"></a>图</h1><h2 id="分类"><a href="#分类" class="headerlink" title="分类"></a>分类</h2><ol>
<li>无向图(i,j)<br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-28_20-28-45.png"></li>
<li>有向图&lt;i,j&gt;<br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-28_20-29-34.png"><br>边数 = 度数之和/2</li>
</ol>
<h2 id="完全图"><a href="#完全图" class="headerlink" title="完全图"></a>完全图</h2><p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-28_20-31-39.png">  </p>
<h2 id="子图"><a href="#子图" class="headerlink" title="子图"></a>子图</h2><h3 id="概念："><a href="#概念：" class="headerlink" title="概念："></a>概念：</h3><p>设有两个图G=(V，E)和G’=(V’，E’)，若V’是V的子集，且E’是E的子集，则称G’是G的子图。<br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-28_20-34-42.png"></p>
<h2 id="路径和路径长度"><a href="#路径和路径长度" class="headerlink" title="路径和路径长度"></a>路径和路径长度</h2><h3 id="路径-在一个图G-V，E-中，从顶点i到顶点j的一条路径-i，i1，i2，…，im，j-。"><a href="#路径-在一个图G-V，E-中，从顶点i到顶点j的一条路径-i，i1，i2，…，im，j-。" class="headerlink" title="路径: 在一个图G=(V，E)中，从顶点i到顶点j的一条路径(i，i1，i2，…，im，j)。"></a>路径: 在一个图G=(V，E)中，从顶点i到顶点j的一条路径(i，i1，i2，…，im，j)。</h3><h3 id="路径长度-是指一条路径上经过的边的数目"><a href="#路径长度-是指一条路径上经过的边的数目" class="headerlink" title="路径长度: 是指一条路径上经过的边的数目"></a>路径长度: 是指一条路径上经过的边的数目</h3><h3 id="简单路径-若一条路径上除开始点和结束点可以相同外，其余顶点均不相同，则称此路径为简单路径"><a href="#简单路径-若一条路径上除开始点和结束点可以相同外，其余顶点均不相同，则称此路径为简单路径" class="headerlink" title="简单路径: 若一条路径上除开始点和结束点可以相同外，其余顶点均不相同，则称此路径为简单路径"></a>简单路径: 若一条路径上除开始点和结束点可以相同外，其余顶点均不相同，则称此路径为简单路径</h3><blockquote>
<p>0 → 1的一条简单路径</p>
<blockquote>
<p>长度为2</p>
</blockquote>
</blockquote>
<h2 id="连通、连通图和连通分量（无向图）"><a href="#连通、连通图和连通分量（无向图）" class="headerlink" title="连通、连通图和连通分量（无向图）"></a>连通、连通图和连通分量（无向图）</h2><h3 id="连通：若从顶点i到顶点j有路径，则称顶点i和j是连通的"><a href="#连通：若从顶点i到顶点j有路径，则称顶点i和j是连通的" class="headerlink" title="连通：若从顶点i到顶点j有路径，则称顶点i和j是连通的"></a>连通：若从顶点i到顶点j有路径，则称顶点i和j是连通的</h3><h3 id="连通图：若图中任意两个顶点都连通，则称为连通图，否则称为非连通图"><a href="#连通图：若图中任意两个顶点都连通，则称为连通图，否则称为非连通图" class="headerlink" title="连通图：若图中任意两个顶点都连通，则称为连通图，否则称为非连通图"></a>连通图：若图中<strong>任意两个顶点都连通</strong>，则称为连通图，否则称为非连通图</h3><h3 id="连通分量：无向图G中的极大连通子图称为G的连通分量。显然，任何连通图的连通分量只有一个，即本身，而非连通图有多个连通分量"><a href="#连通分量：无向图G中的极大连通子图称为G的连通分量。显然，任何连通图的连通分量只有一个，即本身，而非连通图有多个连通分量" class="headerlink" title="连通分量：无向图G中的极大连通子图称为G的连通分量。显然，任何连通图的连通分量只有一个，即本身，而非连通图有多个连通分量"></a>连通分量：无向图G中的极大连通子图称为G的连通分量。显然，任何连通图的连通分量只有一个，即本身，而非连通图有多个连通分量</h3><p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-28_20-47-57.png"></p>
<h2 id="强连通图和强连通分量（有向图）"><a href="#强连通图和强连通分量（有向图）" class="headerlink" title="强连通图和强连通分量（有向图）"></a>强连通图和强连通分量（有向图）</h2><h3 id="强连通图：若图G中的任意两个顶点i和j都连通，即从顶点i到j和从顶点j到i都存在路径，则称图G是强连通图"><a href="#强连通图：若图G中的任意两个顶点i和j都连通，即从顶点i到j和从顶点j到i都存在路径，则称图G是强连通图" class="headerlink" title="强连通图：若图G中的任意两个顶点i和j都连通，即从顶点i到j和从顶点j到i都存在路径，则称图G是强连通图"></a>强连通图：若图G中的任意两个顶点i和j都连通，即从顶点i到j和从顶点j到i都存在路径，则称图G是强连通图</h3><h3 id="强连通分量：有向图G中的极大强连通子图称为G的强连通分量。显然，强连通图只有一个强连通分量，即本身。非强连通图有多个强连通分量"><a href="#强连通分量：有向图G中的极大强连通子图称为G的强连通分量。显然，强连通图只有一个强连通分量，即本身。非强连通图有多个强连通分量" class="headerlink" title="强连通分量：有向图G中的极大强连通子图称为G的强连通分量。显然，强连通图只有一个强连通分量，即本身。非强连通图有多个强连通分量"></a>强连通分量：有向图G中的极大强连通子图称为G的强连通分量。显然，强连通图只有一个强连通分量，即本身。非强连通图有多个强连通分量</h3><p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-28_20-47-18.png"></p>
<h3 id="找强连通分量的方法"><a href="#找强连通分量的方法" class="headerlink" title="找强连通分量的方法"></a>找强连通分量的方法</h3><ol>
<li>找有向环</li>
<li>扩展该有向环：如果某个顶点到该环中任一顶点有路径，并且该环中任一顶点到这个顶点也有路径，则加入这个顶点</li>
</ol>
<p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-28_20-50-13.png"></p>
<h2 id="图的存储"><a href="#图的存储" class="headerlink" title="图的存储"></a>图的存储</h2><h3 id="邻接矩阵（唯一）"><a href="#邻接矩阵（唯一）" class="headerlink" title="邻接矩阵（唯一）"></a>邻接矩阵（唯一）</h3><ol>
<li>无向图<br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-30_15-59-00.png"><br>矩阵(对称)<br>{<br> 0 , 1 , 0 , 1 , 1 ,<br> 1 , 0 , 1 , 1 , 0 ,<br> 0 , 1 , 0 , 1 , 1 ,<br> 1 , 1 , 1 , 0 , 1 ,<br> 1 , 0 , 1 , 1 , 0<br>}</li>
<li>有向图<br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-30_16-04-47.png"><br>矩阵(不一定对称)<br>{<br> 0 , 1 , 0 , 1 , 0 ,<br> 0 , 0 , 1 , 1 , 0 ,<br> 0 , 0 , 0 , 1 , 1 ,<br> 0 , 0 , 0 , 0 , 0 ,<br> 1 , 0 , 0 , 1 , 0<br>}</li>
</ol>
<h4 id="特点-2"><a href="#特点-2" class="headerlink" title="特点"></a>特点</h4><ol>
<li>唯一确定</li>
<li>适合稠密图，空间复杂度为O(n^2)<h4 id="代码实现-10"><a href="#代码实现-10" class="headerlink" title="代码实现"></a>代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">define</span> MaxSize 100</span></span><br><span class="line"><span class="keyword">typedef</span> <span class="class"><span class="keyword">struct</span></span></span><br><span class="line"><span class="class">&#123;</span></span><br><span class="line">    <span class="keyword">int</span> number;</span><br><span class="line">    InfoType info;<span class="comment">//顶点其他信息</span></span><br><span class="line">&#125;Vertex;<span class="comment">//vertex:[几何]顶点</span></span><br><span class="line"><span class="keyword">typedef</span> <span class="class"><span class="keyword">struct</span></span></span><br><span class="line"><span class="class">&#123;</span></span><br><span class="line">    <span class="keyword">int</span> adjacency_matrix[MaxSize][MaxSize];<span class="comment">//邻接矩阵</span></span><br><span class="line">    <span class="keyword">int</span> node,edge;<span class="comment">//顶点数，边数</span></span><br><span class="line">    Vertex v[MaxSize];</span><br><span class="line">&#125;MatGraph;</span><br></pre></td></tr></table></figure></li>
</ol>
<h3 id="邻接表"><a href="#邻接表" class="headerlink" title="邻接表"></a>邻接表</h3><h4 id="存储方式"><a href="#存储方式" class="headerlink" title="存储方式"></a>存储方式</h4><p><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-04-30_16-21-23.png"><br>使用顺序分配和链式分配相结合</p>
<ol>
<li>表头(v1~vn)使用顺序存储</li>
<li>节点使用链式存储</li>
</ol>
<h4 id="特点-3"><a href="#特点-3" class="headerlink" title="特点"></a>特点</h4><ol>
<li>邻接表不唯一</li>
<li>适合稀疏图，空间复杂度O(n+e)</li>
</ol>
<h4 id="代码实现-11"><a href="#代码实现-11" class="headerlink" title="代码实现"></a>代码实现</h4><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">define</span> MaxSize 100</span></span><br><span class="line"><span class="keyword">typedef</span> <span class="class"><span class="keyword">struct</span> <span class="title">AdjacencyNode</span></span></span><br><span class="line"><span class="class">&#123;</span></span><br><span class="line">    <span class="keyword">int</span> adjacency_vertex;</span><br><span class="line">    <span class="keyword">double</span> weight;</span><br><span class="line">    AdjacencyNode *next;</span><br><span class="line">&#125;AdjNode;</span><br><span class="line"><span class="keyword">typedef</span> <span class="class"><span class="keyword">struct</span> <span class="title">Vnode</span></span></span><br><span class="line"><span class="class">&#123;</span></span><br><span class="line">    InfoType info;<span class="comment">//顶点其他信息</span></span><br><span class="line">    AdjNode *first_node;<span class="comment">//第一个节点</span></span><br><span class="line">&#125;</span><br><span class="line"><span class="keyword">typedef</span> <span class="class"><span class="keyword">struct</span> </span></span><br><span class="line"><span class="class">&#123;</span></span><br><span class="line">   Vnode  adjlist[MaxSize];	<span class="comment">//邻接表的表头数组</span></span><br><span class="line">   <span class="keyword">int</span> n，e;<span class="comment">//图中顶点数n和边数e</span></span><br><span class="line">&#125; AdjGraph;</span><br><span class="line"></span><br></pre></td></tr></table></figure>
<h3 id="图的基本运算"><a href="#图的基本运算" class="headerlink" title="图的基本运算"></a>图的基本运算</h3><h4 id="创建-邻接矩阵-gt-邻接表"><a href="#创建-邻接矩阵-gt-邻接表" class="headerlink" title="创建(邻接矩阵-&gt;邻接表)"></a>创建(邻接矩阵-&gt;邻接表)</h4><p>代码实现</p>
<figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">CreateAdjacency</span><span class="params">(AdjGraph *&amp;G,<span class="keyword">int</span> A[MaxSize][MaxSize],<span class="keyword">int</span> n,<span class="keyword">int</span> e)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    AdjNode *p;</span><br><span class="line">    G = (AdjGraph *)<span class="built_in">malloc</span>(<span class="built_in"><span class="keyword">sizeof</span></span>(AdjGraph));</span><br><span class="line">    <span class="keyword">int</span> i,j;</span><br><span class="line">    <span class="keyword">for</span>(i = <span class="number">0</span>;i&lt;n;i++)</span><br><span class="line">    &#123;</span><br><span class="line">        G-&gt;adjlist[i].first_node = <span class="literal">NULL</span>;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">for</span>(i = <span class="number">0</span>;i&lt;n;i++)</span><br><span class="line">        <span class="keyword">for</span>(j = n<span class="number">-1</span>;j&gt;=<span class="number">0</span>;j--)</span><br><span class="line">        <span class="keyword">if</span>(A[i][j] != <span class="number">0</span> &amp;&amp; A[i][j] != INF)</span><br><span class="line">        &#123;</span><br><span class="line">            p = (AdjNode *)<span class="built_in">malloc</span>(<span class="built_in"><span class="keyword">sizeof</span></span>(AdjNode));</span><br><span class="line">            p-&gt;adjacency_vertex = j;</span><br><span class="line">            p-&gt;weight = A[i][j];</span><br><span class="line">            p-&gt;next = G-&gt;adjlist[i].first_node;<span class="comment">//头插法</span></span><br><span class="line">            G-&gt;adjlist[i].first_node = p;</span><br><span class="line">        &#125;</span><br><span class="line">        G-&gt;n = n;</span><br><span class="line">        G-&gt;e = e;</span><br><span class="line">&#125;<span class="comment">//时间复杂度O(n^2)</span></span><br></pre></td></tr></table></figure>
<h4 id="创建-邻接表-gt-邻接矩阵"><a href="#创建-邻接表-gt-邻接矩阵" class="headerlink" title="创建(邻接表-&gt;邻接矩阵)"></a>创建(邻接表-&gt;邻接矩阵)</h4><p>代码实现</p>
<figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">List2Map</span><span class="params">(AdjGraph *G,MatGraph &amp;m)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">int</span> i;</span><br><span class="line">    AdjNode *p;</span><br><span class="line">    <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i &lt; G-&gt;n; i++)</span><br><span class="line">    &#123;</span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>; j &lt; G-&gt;n; j++)</span><br><span class="line">        &#123;</span><br><span class="line">            m.adjacency_matrix[i][j] = <span class="number">0</span>;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">for</span>(i = <span class="number">0</span>;i &lt; G-&gt;n;i++)</span><br><span class="line">    &#123;</span><br><span class="line">        p = G-&gt;adjlist[i].first_node;</span><br><span class="line">        <span class="keyword">while</span>(p!=<span class="literal">NULL</span>)</span><br><span class="line">        &#123;</span><br><span class="line">            m.adjacency_matrix[i][p-&gt;adjacency_vertex] = p-&gt;weight;</span><br><span class="line">            p = p-&gt;next;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">    m.edge = G-&gt;e;</span><br><span class="line">    m.node = G-&gt;n;</span><br><span class="line">&#125;<span class="comment">//时间复杂度O(n+e)</span></span><br></pre></td></tr></table></figure>

<h3 id="展示图的邻接表"><a href="#展示图的邻接表" class="headerlink" title="展示图的邻接表"></a>展示图的邻接表</h3><p>代码实现</p>
<figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">DisplayGraph</span><span class="params">(AdjGraph *G)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">int</span> i;</span><br><span class="line">    AdjNode *p;</span><br><span class="line">    <span class="keyword">for</span>(i = <span class="number">0</span>;i &lt; G-&gt;n;i++)</span><br><span class="line">    &#123;</span><br><span class="line">        p = G-&gt;adjlist[i].first_node;</span><br><span class="line">        cout&lt;&lt;i&lt;&lt;<span class="string">&quot; : &quot;</span>;</span><br><span class="line">        <span class="keyword">while</span>(p != <span class="literal">NULL</span>)</span><br><span class="line">        &#123;</span><br><span class="line">            cout&lt;&lt;p-&gt;adjacency_vertex&lt;&lt;<span class="string">&quot;[&quot;</span>&lt;&lt;p-&gt;weight&lt;&lt;<span class="string">&quot;] -&gt; &quot;</span>;</span><br><span class="line">            p = p-&gt;next;</span><br><span class="line">        &#125;</span><br><span class="line">        cout&lt;&lt;<span class="string">&quot;∧&quot;</span>&lt;&lt;endl;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<h3 id="图的销毁"><a href="#图的销毁" class="headerlink" title="图的销毁"></a>图的销毁</h3><p>代码实现</p>
<figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">DestroyGraph</span><span class="params">(AdjGraph *&amp;G)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">int</span> i;</span><br><span class="line">    AdjNode *p,*pre;</span><br><span class="line">    <span class="keyword">for</span>(i = <span class="number">0</span>;i &lt; G-&gt;n;i++)</span><br><span class="line">    &#123;</span><br><span class="line">        pre = G-&gt;adjlist[i].first_node;</span><br><span class="line">        <span class="keyword">if</span>(pre!=<span class="literal">NULL</span>)</span><br><span class="line">        &#123;</span><br><span class="line">            p = pre-&gt;next;</span><br><span class="line">            <span class="keyword">while</span>(p != <span class="literal">NULL</span>)</span><br><span class="line">            &#123;</span><br><span class="line">                <span class="built_in">free</span>(pre);</span><br><span class="line">                pre = p;</span><br><span class="line">                p = p-&gt;next;</span><br><span class="line">            &#125;</span><br><span class="line">            <span class="built_in">free</span>(pre);</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="built_in">free</span>(G);<span class="comment">//释放头节点数组</span></span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<h3 id="其他储存方法"><a href="#其他储存方法" class="headerlink" title="其他储存方法"></a>其他储存方法</h3><h4 id="十字链表"><a href="#十字链表" class="headerlink" title="十字链表"></a>十字链表</h4><p>数据结构<br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-05-01_22-40-58.png"><br>结构<br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-05-01_22-42-37.png"></p>
<h4 id="邻接多重表"><a href="#邻接多重表" class="headerlink" title="邻接多重表"></a>邻接多重表</h4><p>数据结构<br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-05-01_22-43-34.png"><br>结构<br><img src="/2021/05/04/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/Snipaste_2021-05-01_22-45-01.png"></p>

      

      
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<ul>
<li><a href="#%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C">计算机网络</a><ul>
<li><a href="#%E6%A6%82%E5%BF%B5">概念</a></li>
<li><a href="#%E5%8A%9F%E8%83%BD">功能</a></li>
<li><a href="#%E7%BB%84%E6%88%90">组成</a><ul>
<li><a href="#%E7%BB%84%E6%88%90%E9%83%A8%E5%88%86">组成部分</a></li>
<li><a href="#%E5%B7%A5%E4%BD%9C%E6%96%B9%E5%BC%8F">工作方式</a></li>
<li><a href="#%E5%8A%9F%E8%83%BD%E7%BB%84%E6%88%90">功能组成</a></li>
</ul>
</li>
<li><a href="#%E5%88%86%E7%B1%BB">分类</a><ul>
<li><a href="#%E5%88%86%E5%B8%83%E8%8C%83%E5%9B%B4">分布范围</a></li>
<li><a href="#%E4%BD%BF%E7%94%A8%E8%80%85">使用者</a></li>
<li><a href="#%E4%BA%A4%E6%8D%A2%E6%8A%80%E6%9C%AF">交换技术</a></li>
<li><a href="#%E6%8B%93%E6%89%91%E7%BB%93%E6%9E%84">拓扑结构</a></li>
<li><a href="#%E4%BC%A0%E8%BE%93%E6%8A%80%E6%9C%AF">传输技术</a></li>
<li><a href="#%E7%9F%A5%E8%AF%86%E6%A1%86%E6%9E%B6">知识框架</a></li>
</ul>
</li>
<li><a href="#%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C%E6%A0%87%E5%87%86%E5%8C%96">计算机网络标准化</a><ul>
<li><a href="#%E6%A0%87%E5%87%86%E5%88%86%E7%B1%BB">标准分类</a></li>
<li><a href="#rfcrequest-for-comments---%E5%9B%A0%E7%89%B9%E7%BD%91%E6%A0%87%E5%87%86%E7%9A%84%E5%BD%A2%E5%BC%8F">RFC(Request For Comments)—因特网标准的形式</a><ul>
<li><a href="#rfc%E4%B8%8A%E5%8D%87%E4%B8%BA%E5%9B%A0%E7%89%B9%E7%BD%91%E6%AD%A3%E5%BC%8F%E6%A0%87%E5%87%86%E7%9A%84%E5%9B%9B%E4%B8%AA%E9%98%B6%E6%AE%B5">RFC上升为因特网正式标准的四个阶段</a></li>
<li><a href="#%E7%9B%B8%E5%85%B3%E7%BB%84%E7%BB%87">相关组织</a></li>
</ul>
</li>
<li><a href="#%E7%9F%A5%E8%AF%86%E6%A1%86%E6%9E%B6-1">知识框架</a></li>
</ul>
</li>
<li><a href="#%E6%80%A7%E8%83%BD%E6%8C%87%E6%A0%87">性能指标</a><ul>
<li><a href="#%E9%80%9F%E7%8E%87">速率</a><ul>
<li><a href="#%E5%88%AB%E7%A7%B0%E6%95%B0%E6%8D%AE%E7%8E%87%E6%95%B0%E6%8D%AE%E4%BC%A0%E8%BE%93%E7%8E%87%E6%AF%94%E7%89%B9%E7%8E%87%E6%AF%94%E7%89%B910--%E4%BD%8D">别称：<strong>数据率</strong>，<strong>数据传输率</strong>，<strong>比特率</strong>(比特：1/0  位)</a></li>
<li><a href="#%E6%A6%82%E5%BF%B5">概念：</a></li>
<li><a href="#%E5%8D%95%E4%BD%8D">单位：</a></li>
</ul>
</li>
<li><a href="#%E5%B8%A6%E5%AE%BD">带宽</a><ul>
<li><a href="#%E6%A6%82%E5%BF%B5-1">概念：</a></li>
</ul>
</li>
<li><a href="#%E5%90%9E%E5%90%90%E9%87%8F">吞吐量</a><ul>
<li><a href="#%E6%A6%82%E5%BF%B5-2">概念：</a></li>
</ul>
</li>
<li><a href="#%E6%97%B6%E5%BB%B6">时延</a><ul>
<li><a href="#%E6%A6%82%E5%BF%B5-3">概念：</a></li>
<li><a href="#%E5%88%AB%E7%A7%B0">别称：</a></li>
<li><a href="#%E5%88%86%E7%B1%BB-1">分类</a></li>
</ul>
</li>
<li><a href="#%E6%97%B6%E5%BB%B6%E5%B8%A6%E5%AE%BD%E7%A7%AF">时延带宽积</a></li>
<li><a href="#%E5%BE%80%E8%BF%94%E6%97%B6%E5%BB%B6rtt">往返时延RTT</a><ul>
<li><a href="#%E6%A6%82%E5%BF%B5-4">概念：</a></li>
<li><a href="#rtt%E5%8C%85%E6%8B%AC">RTT包括</a></li>
</ul>
</li>
<li><a href="#%E5%88%A9%E7%94%A8%E7%8E%87">利用率</a></li>
<li><a href="#%E7%9F%A5%E8%AF%86%E6%A1%86%E6%9E%B6-2">知识框架</a></li>
</ul>
</li>
<li><a href="#%E5%88%86%E5%B1%82%E7%BB%93%E6%9E%84">分层结构</a><ul>
<li><a href="#%E5%9F%BA%E6%9C%AC%E5%8E%9F%E5%88%99">基本原则</a></li>
<li><a href="#%E7%BB%93%E6%9E%84">结构</a></li>
<li><a href="#%E6%95%B0%E6%8D%AE%E4%BC%A0%E8%BE%93">数据传输</a></li>
</ul>
</li>
<li><a href="#osi">OSI</a><ul>
<li><a href="#%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C%E5%88%86%E5%B1%82%E7%BB%93%E6%9E%84">计算机网络分层结构</a></li>
<li><a href="#osi%E5%BB%BA%E7%AB%8B%E7%9A%84%E7%9B%AE%E7%9A%84">OSI建立的目的</a></li>
<li><a href="#%E7%BB%93%E6%9E%84-1">结构</a></li>
<li><a href="#%E9%80%9A%E4%BF%A1%E8%BF%87%E7%A8%8B">通信过程</a></li>
<li><a href="#%E5%BA%94%E7%94%A8%E5%B1%82">应用层</a><ul>
<li><a href="#%E5%AE%9A%E4%B9%89%E6%89%80%E6%9C%89%E5%92%8C%E7%94%A8%E6%88%B7%E4%BA%A4%E4%BA%92%E4%BA%A7%E7%94%9F%E7%BD%91%E7%BB%9C%E6%B5%81%E9%87%8F%E7%9A%84%E7%A8%8B%E5%BA%8F">定义：所有和用户交互产生网络流量的程序</a></li>
<li><a href="#%E6%9C%8D%E5%8A%A1%E5%8D%8F%E8%AE%AE%E6%96%87%E4%BB%B6%E4%BC%A0%E8%BE%93ftp%E7%94%B5%E5%AD%90%E9%82%AE%E4%BB%B6smtp%E4%B8%87%E7%BB%B4%E7%BD%91http">服务（协议）：文件传输（FTP），电子邮件（SMTP），万维网（HTTP）</a></li>
</ul>
</li>
<li><a href="#%E8%A1%A8%E7%A4%BA%E5%B1%82">表示层</a><ul>
<li><a href="#%E5%8A%9F%E8%83%BD-1">功能</a></li>
</ul>
</li>
<li><a href="#%E4%BC%9A%E8%AF%9D%E5%B1%82">会话层</a><ul>
<li><a href="#%E4%BD%9C%E7%94%A8%E5%90%91%E8%A1%A8%E7%A4%BA%E5%B1%82%E7%94%A8%E6%88%B7%E8%BF%9B%E7%A8%8B%E5%BB%BA%E7%AB%8B%E8%BF%9E%E6%8E%A5%E5%B9%B6%E5%9C%A8%E8%BF%9E%E6%8E%A5%E4%B8%8A%E6%9C%89%E5%BA%8F%E5%9C%B0%E4%BC%A0%E8%BE%93%E6%95%B0%E6%8D%AE%E8%BF%99%E6%98%AF%E4%BC%9A%E8%AF%9D%E4%B9%9F%E6%98%AF%E5%BB%BA%E7%AB%8B%E5%90%8C%E6%AD%A5syn">作用：向表示层/用户进程<strong>建立连接</strong>并在连接上<strong>有序</strong>地<strong>传输</strong>数据，这是会话，也是<strong>建立同步</strong>（SYN）</a></li>
<li><a href="#%E5%8A%9F%E8%83%BD-2">功能</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#tcpip">TCP/IP</a><ul>
<li><a href="#%E6%A8%A1%E5%9E%8B">模型</a></li>
<li><a href="#%E4%B8%8Eosi%E7%9A%84%E7%9B%B8%E5%90%8C%E7%82%B9">与OSI的相同点</a></li>
<li><a href="#%E4%B8%8Eosi%E7%9A%84%E4%B8%8D%E5%90%8C%E7%82%B9">与OSI的不同点</a></li>
</ul>
</li>
<li><a href="#%E4%BA%94%E5%B1%82%E5%8F%82%E8%80%83%E6%A8%A1%E5%9E%8B">五层参考模型</a><ul>
<li><a href="#%E6%A8%A1%E5%9E%8B%E7%BB%BC%E5%90%88osi%E5%92%8Ctcpip%E7%9A%84%E4%BC%98%E7%82%B9">模型（综合OSI和TCP/IP的优点）</a></li>
<li><a href="#%E5%B0%81%E8%A3%85%E5%92%8C%E8%A7%A3%E5%B0%81%E8%A3%85">封装和解封装</a></li>
</ul>
</li>
<li><a href="#%E7%AC%AC%E4%B8%80%E7%AB%A0%E6%80%BB%E7%BB%93">第一章总结</a></li>
<li><a href="#%E7%89%A9%E7%90%86%E5%B1%82">物理层</a><ul>
<li><a href="#%E6%A6%82%E5%BF%B5%E8%A7%A3%E5%86%B3%E5%A6%82%E4%BD%95%E5%9C%A8%E8%BF%9E%E6%8E%A5%E5%90%84%E7%A7%8D%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%9A%84%E4%BC%A0%E8%BE%93%E5%AA%92%E4%BD%93%E4%B8%8A%E4%BC%A0%E8%BE%93%E6%95%B0%E6%8D%AE%E6%AF%94%E7%89%B9%E6%B5%81%E8%80%8C%E4%B8%8D%E6%98%AF%E5%85%B7%E4%BD%93%E7%9A%84%E4%BC%A0%E8%BE%93%E5%AA%92%E4%BD%93">概念：解决如何在连接各种计算机的传输媒体上<strong>传输数据比特流</strong>，而不是具体的传输媒体</a></li>
<li><a href="#%E4%BB%BB%E5%8A%A1%E7%A1%AE%E5%AE%9A%E4%B8%8E%E4%BC%A0%E8%BE%93%E5%AA%92%E4%BD%93%E6%8E%A5%E5%8F%A3%E6%9C%89%E5%85%B3%E7%9A%84%E4%B8%80%E4%BA%9B%E7%89%B9%E6%80%A7---%E5%AE%9A%E4%B9%89%E6%A0%87%E5%87%86">任务：确定与传输媒体<strong>接口</strong>有关的一些特性 -&gt; 定义标准</a></li>
<li><a href="#%E7%89%B9%E6%80%A7">特性</a></li>
</ul>
</li>
<li><a href="#%E6%95%B0%E6%8D%AE%E9%80%9A%E4%BF%A1">数据通信</a><ul>
<li><a href="#%E6%A8%A1%E5%9E%8B-1">模型</a></li>
<li><a href="#%E7%9B%AE%E7%9A%84%E4%BC%A0%E9%80%81%E6%B6%88%E6%81%AF">目的：传送消息</a></li>
<li><a href="#%E6%A6%82%E5%BF%B5-1">概念</a></li>
<li><a href="#%E9%80%9A%E4%BF%A1%E6%96%B9%E5%BC%8F">通信方式</a></li>
<li><a href="#%E6%95%B0%E6%8D%AE%E4%BC%A0%E8%BE%93%E6%96%B9%E5%BC%8F">数据传输方式</a></li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- /TOC -->
<h1 id="计算机网络"><a href="#计算机网络" class="headerlink" title="计算机网络"></a>计算机网络</h1><h2 id="概念"><a href="#概念" class="headerlink" title="概念"></a>概念</h2><p>是一个将分散的，有独立功能的<strong>计算机系统</strong>，通过<strong>通信设备（交换机，路由器）</strong> 和 <strong>线路</strong>连接起来，由功能完善的<strong>软件</strong>实现<strong>资源共享</strong>和<strong>信息传递</strong>的系统。<br>是<strong>互联的-互联互通</strong>，<strong>自治的-无主从关系</strong>计算机集合。</p>
<h2 id="功能"><a href="#功能" class="headerlink" title="功能"></a>功能</h2><ol>
<li><strong>数据通信</strong></li>
<li><strong>资源共享</strong></li>
<li>分布式处理</li>
<li>提高可靠性</li>
<li>负载均衡</li>
</ol>
<h2 id="组成"><a href="#组成" class="headerlink" title="组成"></a>组成</h2><h3 id="组成部分"><a href="#组成部分" class="headerlink" title="组成部分"></a>组成部分</h3><ol>
<li>软件</li>
<li>硬件</li>
<li>协议</li>
</ol>
<h3 id="工作方式"><a href="#工作方式" class="headerlink" title="工作方式"></a>工作方式</h3><ol>
<li>边缘部分：用户直接使用<br> C/S方式<br> P2P方式</li>
<li>核心部分：为边缘服务</li>
</ol>
<h3 id="功能组成"><a href="#功能组成" class="headerlink" title="功能组成"></a>功能组成</h3><ol>
<li>通信子网：实现<strong>数据通信</strong></li>
<li>资源子网：实现<strong>资源共享</strong>/数据处理</li>
</ol>
<p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-16_10-48-10.png"></p>
<h2 id="分类"><a href="#分类" class="headerlink" title="分类"></a>分类</h2><h3 id="分布范围"><a href="#分布范围" class="headerlink" title="分布范围"></a>分布范围</h3><ol>
<li>广域网WAN</li>
<li>城域网MAN</li>
<li>局域网LAN</li>
<li>个人区域网PAN</li>
</ol>
<h3 id="使用者"><a href="#使用者" class="headerlink" title="使用者"></a>使用者</h3><ol>
<li>公用网</li>
<li>专用网</li>
</ol>
<h3 id="交换技术"><a href="#交换技术" class="headerlink" title="交换技术"></a>交换技术</h3><ol>
<li>电路交换</li>
<li>报文交换</li>
<li>分组交换</li>
</ol>
<h3 id="拓扑结构"><a href="#拓扑结构" class="headerlink" title="拓扑结构"></a>拓扑结构</h3><ol>
<li>总线型</li>
<li>星型</li>
<li>环型</li>
<li>网状型（广域网）</li>
</ol>
<p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-16_11-02-25.png"></p>
<h3 id="传输技术"><a href="#传输技术" class="headerlink" title="传输技术"></a>传输技术</h3><ol>
<li>广播式网络：共享公共通信信道</li>
<li>点对点网络：使用<strong>分组存储转发</strong>和<strong>路由选择</strong>机制</li>
</ol>
<h3 id="知识框架"><a href="#知识框架" class="headerlink" title="知识框架"></a>知识框架</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-16_11-12-04.png"></p>
<hr>
<h2 id="计算机网络标准化"><a href="#计算机网络标准化" class="headerlink" title="计算机网络标准化"></a>计算机网络标准化</h2><h3 id="标准分类"><a href="#标准分类" class="headerlink" title="标准分类"></a>标准分类</h3><ol>
<li>法定标准：OSI</li>
<li>事实标准：TCP/IP</li>
</ol>
<h3 id="RFC-Request-For-Comments-—因特网标准的形式"><a href="#RFC-Request-For-Comments-—因特网标准的形式" class="headerlink" title="RFC(Request For Comments)—因特网标准的形式"></a>RFC(Request For Comments)—因特网标准的形式</h3><h4 id="RFC上升为因特网正式标准的四个阶段"><a href="#RFC上升为因特网正式标准的四个阶段" class="headerlink" title="RFC上升为因特网正式标准的四个阶段"></a>RFC上升为因特网正式标准的四个阶段</h4><ol>
<li>因特网草案</li>
<li>建议标准</li>
<li>因特网标准</li>
</ol>
<h4 id="相关组织"><a href="#相关组织" class="headerlink" title="相关组织"></a>相关组织</h4><ol>
<li>ISO 国际标准化组织</li>
<li>ITU 国际电信联盟</li>
<li>IEEE 电气电子工程师学会</li>
<li>IETF Internet工程任务组</li>
</ol>
<h3 id="知识框架-1"><a href="#知识框架-1" class="headerlink" title="知识框架"></a>知识框架</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-17_19-46-42.png"></p>
<hr>
<h2 id="性能指标"><a href="#性能指标" class="headerlink" title="性能指标"></a>性能指标</h2><h3 id="速率"><a href="#速率" class="headerlink" title="速率"></a>速率</h3><h4 id="别称：数据率，数据传输率，比特率-比特：1-0-位"><a href="#别称：数据率，数据传输率，比特率-比特：1-0-位" class="headerlink" title="别称：数据率，数据传输率，比特率(比特：1/0  位)"></a>别称：<strong>数据率</strong>，<strong>数据传输率</strong>，<strong>比特率</strong>(比特：1/0  位)</h4><h4 id="概念："><a href="#概念：" class="headerlink" title="概念："></a>概念：</h4><p>连接在计算机网络上的<strong>主机</strong>在数字信道上传输数据<strong>位数的速率</strong></p>
<h4 id="单位："><a href="#单位：" class="headerlink" title="单位："></a>单位：</h4><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-17_19-58-28.png"></p>
<h3 id="带宽"><a href="#带宽" class="headerlink" title="带宽"></a>带宽</h3><h4 id="概念：-1"><a href="#概念：-1" class="headerlink" title="概念："></a>概念：</h4><p>网络的通信线路传输数据的能力，通常是指单位时间内从网络中一点到另一点能通过的<strong>最高数据率</strong>，单位同速率。<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-17_20-06-57.png"></p>
<h3 id="吞吐量"><a href="#吞吐量" class="headerlink" title="吞吐量"></a>吞吐量</h3><h4 id="概念：-2"><a href="#概念：-2" class="headerlink" title="概念："></a>概念：</h4><p>单位时间内通过某个 <strong>网络（或者信道，接口）</strong> 的数据量，单位同速率。<br>吞吐量受<strong>网络带宽</strong>或<strong>网络额定速率</strong>的限制<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-17_20-14-26.png"></p>
<h3 id="时延"><a href="#时延" class="headerlink" title="时延"></a>时延</h3><h4 id="概念：-3"><a href="#概念：-3" class="headerlink" title="概念："></a>概念：</h4><p>指数据（报文/分组/比特流）从网络（或链路）的一端传输到另一端所需的时间</p>
<h4 id="别称："><a href="#别称：" class="headerlink" title="别称："></a>别称：</h4><p>延迟，迟延</p>
<h4 id="分类-1"><a href="#分类-1" class="headerlink" title="分类"></a>分类</h4><ol>
<li>发送时延（传输时延）<br>发送时延 = 数据长度/信道带宽（发送速率）<br>高速链路</li>
<li>传播时延<br>传播时延 = 信道长度/电磁波在信道上的传播速率</li>
<li>排队时延</li>
<li>处理时延</li>
</ol>
<h3 id="时延带宽积"><a href="#时延带宽积" class="headerlink" title="时延带宽积"></a>时延带宽积</h3><p>时延带宽积 = 传播时延 * 带宽<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-17_20-29-58.png"></p>
<h3 id="往返时延RTT"><a href="#往返时延RTT" class="headerlink" title="往返时延RTT"></a>往返时延RTT</h3><h4 id="概念：-4"><a href="#概念：-4" class="headerlink" title="概念："></a>概念：</h4><p>从发送方发送数据开始，到发送方收到接收方的确认总共经历的时延<br>RTT越大，在收到确认之前可以发送的数据越多</p>
<h4 id="RTT包括"><a href="#RTT包括" class="headerlink" title="RTT包括"></a>RTT包括</h4><p>传播时延*2 + 末端处理时间</p>
<h3 id="利用率"><a href="#利用率" class="headerlink" title="利用率"></a>利用率</h3><ol>
<li>网络利用率 = 信道利用率加权平均值</li>
<li>信道利用率 = 有数据通过时间/（有+无）数据通过时间</li>
</ol>
<h3 id="知识框架-2"><a href="#知识框架-2" class="headerlink" title="知识框架"></a>知识框架</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-17_20-39-56.png"></p>
<hr>
<h2 id="分层结构"><a href="#分层结构" class="headerlink" title="分层结构"></a>分层结构</h2><h3 id="基本原则"><a href="#基本原则" class="headerlink" title="基本原则"></a>基本原则</h3><ol>
<li>各层相互<strong>独立</strong></li>
<li>每层之间<strong>界限自然清晰</strong>，易于理解，相互交流尽可能少</li>
<li>可分割，用<strong>最合适的技术</strong>实现</li>
<li>上下层独立，<strong>上层单向使用下层服务</strong></li>
<li>能促进标准化工作</li>
</ol>
<h3 id="结构"><a href="#结构" class="headerlink" title="结构"></a>结构</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-21_08-42-19.png"></p>
<ol>
<li>实体：n层中活动的元素称为<strong>n层实体</strong>，同一层的称为<strong>对等实体</strong></li>
<li>协议：为网络中<strong>对等实体</strong>数据交换而建立的规则，标准，或者约定成为网络协议[水平]</li>
<li>接口（访问服务点SAP）：上层使用下层的入口</li>
<li>服务：下层为上层提供的功能调用[垂直]</li>
</ol>
<h3 id="数据传输"><a href="#数据传输" class="headerlink" title="数据传输"></a>数据传输</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-21_08-24-41.png"></p>
<ol>
<li>SDU数据服务单元：应传输的数据</li>
<li>PCI协议控制信息：控制协议操作的信息</li>
<li>PDU协议数据单元：对等层传输的数据</li>
</ol>
<hr>
<h2 id="OSI"><a href="#OSI" class="headerlink" title="OSI"></a>OSI</h2><h3 id="计算机网络分层结构"><a href="#计算机网络分层结构" class="headerlink" title="计算机网络分层结构"></a>计算机网络分层结构</h3><ol>
<li>法定标准（7层）：OSI</li>
<li>事实标准（4层）：TCP/IP</li>
</ol>
<h3 id="OSI建立的目的"><a href="#OSI建立的目的" class="headerlink" title="OSI建立的目的"></a>OSI建立的目的</h3><p>支持<strong>异构网络系统</strong>的互联互通</p>
<h3 id="结构-1"><a href="#结构-1" class="headerlink" title="结构"></a>结构</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-21_09-08-26.png"></p>
<h3 id="通信过程"><a href="#通信过程" class="headerlink" title="通信过程"></a>通信过程</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-21_09-15-17.png"><br>上四层端到端，下三层点到点</p>
<h3 id="应用层"><a href="#应用层" class="headerlink" title="应用层"></a>应用层</h3><h4 id="定义：所有和用户交互产生网络流量的程序"><a href="#定义：所有和用户交互产生网络流量的程序" class="headerlink" title="定义：所有和用户交互产生网络流量的程序"></a>定义：所有和用户交互产生网络流量的程序</h4><h4 id="服务（协议）：文件传输（FTP），电子邮件（SMTP），万维网（HTTP）"><a href="#服务（协议）：文件传输（FTP），电子邮件（SMTP），万维网（HTTP）" class="headerlink" title="服务（协议）：文件传输（FTP），电子邮件（SMTP），万维网（HTTP）"></a>服务（协议）：文件传输（FTP），电子邮件（SMTP），万维网（HTTP）</h4><h3 id="表示层"><a href="#表示层" class="headerlink" title="表示层"></a>表示层</h3><h4 id="功能-1"><a href="#功能-1" class="headerlink" title="功能"></a>功能</h4><ol>
<li>数据格式变换</li>
<li>数据加密解密</li>
<li>数据的压缩和恢复</li>
</ol>
<h3 id="会话层"><a href="#会话层" class="headerlink" title="会话层"></a>会话层</h3><h4 id="作用：向表示层-用户进程建立连接并在连接上有序地传输数据，这是会话，也是建立同步（SYN）"><a href="#作用：向表示层-用户进程建立连接并在连接上有序地传输数据，这是会话，也是建立同步（SYN）" class="headerlink" title="作用：向表示层/用户进程建立连接并在连接上有序地传输数据，这是会话，也是建立同步（SYN）"></a>作用：向表示层/用户进程<strong>建立连接</strong>并在连接上<strong>有序</strong>地<strong>传输</strong>数据，这是会话，也是<strong>建立同步</strong>（SYN）</h4><h4 id="功能-2"><a href="#功能-2" class="headerlink" title="功能"></a>功能</h4><ol>
<li>建立，管理，终止会话</li>
<li>会话在通信失效时从<strong>校验点/同步点</strong>继续恢复通信，实现数据同步</li>
</ol>
<hr>
<h2 id="TCP-IP"><a href="#TCP-IP" class="headerlink" title="TCP/IP"></a>TCP/IP</h2><h3 id="模型"><a href="#模型" class="headerlink" title="模型"></a>模型</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-27_14-09-02.png"></p>
<h3 id="与OSI的相同点"><a href="#与OSI的相同点" class="headerlink" title="与OSI的相同点"></a>与OSI的相同点</h3><ol>
<li>分层</li>
<li>基于独立的协议栈的概念</li>
<li>可以实现异构网络的互联</li>
</ol>
<h3 id="与OSI的不同点"><a href="#与OSI的不同点" class="headerlink" title="与OSI的不同点"></a>与OSI的不同点</h3><ol>
<li>OSI定义三点：服务，协议，接口</li>
<li>OSI参考模型先于协议发明，不偏向特定协议</li>
<li>TCP/IP将IP设为重要层面</li>
<li><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-27_14-18-07.png"></li>
</ol>
<hr>
<h2 id="五层参考模型"><a href="#五层参考模型" class="headerlink" title="五层参考模型"></a>五层参考模型</h2><h3 id="模型（综合OSI和TCP-IP的优点）"><a href="#模型（综合OSI和TCP-IP的优点）" class="headerlink" title="模型（综合OSI和TCP/IP的优点）"></a>模型（综合OSI和TCP/IP的优点）</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-27_14-21-34.png"></p>
<h3 id="封装和解封装"><a href="#封装和解封装" class="headerlink" title="封装和解封装"></a>封装和解封装</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-27_14-24-01.png"></p>
<hr>
<h2 id="第一章总结"><a href="#第一章总结" class="headerlink" title="第一章总结"></a>第一章总结</h2><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-27_14-25-56.png"> </p>
<hr>
<h2 id="物理层"><a href="#物理层" class="headerlink" title="物理层"></a>物理层</h2><h3 id="概念：解决如何在连接各种计算机的传输媒体上传输数据比特流，而不是具体的传输媒体"><a href="#概念：解决如何在连接各种计算机的传输媒体上传输数据比特流，而不是具体的传输媒体" class="headerlink" title="概念：解决如何在连接各种计算机的传输媒体上传输数据比特流，而不是具体的传输媒体"></a>概念：解决如何在连接各种计算机的传输媒体上<strong>传输数据比特流</strong>，而不是具体的传输媒体</h3><h3 id="任务：确定与传输媒体接口有关的一些特性-gt-定义标准"><a href="#任务：确定与传输媒体接口有关的一些特性-gt-定义标准" class="headerlink" title="任务：确定与传输媒体接口有关的一些特性 -&gt; 定义标准"></a>任务：确定与传输媒体<strong>接口</strong>有关的一些特性 -&gt; 定义标准</h3><h3 id="特性"><a href="#特性" class="headerlink" title="特性"></a>特性</h3><ol>
<li><p>机械特性：规格，接口形状，<strong>引线数目，引脚数量</strong>，排列情况</p>
</li>
<li><p>电压特性：<strong>电压范围</strong>，阻抗匹配，传输<strong>速率和距离</strong><br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-27_14-37-03.png"></p>
</li>
<li><p>功能特性：某一<strong>电平表示何种意义</strong><br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-27_14-37-12.png"></p>
</li>
<li><p>过程（规程）特性：各条物理线路工作的<strong>规程和时序</strong></p>
</li>
</ol>
<hr>
<h2 id="数据通信"><a href="#数据通信" class="headerlink" title="数据通信"></a>数据通信</h2><h3 id="模型-1"><a href="#模型-1" class="headerlink" title="模型"></a>模型</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-27_14-56-41.png">  </p>
<h3 id="目的：传送消息"><a href="#目的：传送消息" class="headerlink" title="目的：传送消息"></a>目的：传送消息</h3><h3 id="概念-1"><a href="#概念-1" class="headerlink" title="概念"></a>概念</h3><p><strong>数据</strong>是传送消息的实体<br><strong>信号</strong>数据在传输过程中的存在形式</p>
<ol>
<li>数字信号：消息参数的取值是离散的</li>
<li>模拟信号：消息参数的取值是连续的<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-27_14-51-59.png"></li>
</ol>
<p><strong>信源</strong>产生和发送数据的源头<br><strong>信宿</strong>接收数据的终点<br><strong>信道</strong>信号传输的媒介<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-27_14-57-47.png">  </p>
<h3 id="通信方式"><a href="#通信方式" class="headerlink" title="通信方式"></a>通信方式</h3><ol>
<li>单工通信：单方向，只需<strong>一条</strong>信道</li>
<li>半双工通信：双方都可以发送或者接受信息，但是任何一方不能同时发送和接受，需要<strong>两条</strong>信道</li>
<li>全双工通信：双方可以同时接收发送信息，需要<strong>两条</strong>信道</li>
</ol>
<h3 id="数据传输方式"><a href="#数据传输方式" class="headerlink" title="数据传输方式"></a>数据传输方式</h3><ol>
<li>串行传输：<strong>慢，费用低，适合远距离</strong></li>
<li>并行传输：<strong>快，贵，适合近距离</strong><br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BD%91%E7%BB%9C/Snipaste_2021-04-27_15-04-58.png">  </li>
</ol>

      

      
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    <article id="post-计算机组成原理" class="article article-type-post  article-index" itemscope itemprop="blogPost">
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        <a href="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/" class="archive-article-date">
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        <!-- TOC -->

<ul>
<li><a href="#%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86">计算机组成原理</a><ul>
<li><a href="#%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F">计算机系统</a><ul>
<li><a href="#%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F--%E7%A1%AC%E4%BB%B6--%E8%BD%AF%E4%BB%B6-%E8%BD%AF%E4%BB%B6%E5%92%8C%E7%A1%AC%E4%BB%B6%E7%AD%89%E6%95%88">计算机系统 = 硬件 + 软件 (软件和硬件等效)</a><ul>
<li><a href="#%E8%BD%AF%E4%BB%B6">软件</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%A1%AC%E4%BB%B6%E7%9A%84%E5%9F%BA%E6%9C%AC%E7%BB%84%E6%88%90">计算机硬件的基本组成</a><ul>
<li><a href="#%E5%86%AF%E8%AF%BA%E4%BE%9D%E6%9B%BC%E6%9C%BA">冯诺依曼机</a><ul>
<li><a href="#%E6%A6%82%E5%BF%B5%E5%B0%86%E6%8C%87%E4%BB%A4%E4%BB%A5%E4%BA%8C%E8%BF%9B%E5%88%B6%E7%A0%81%E7%9A%84%E5%BD%A2%E5%BC%8F%E5%AE%9E%E7%8E%B0%E8%BE%93%E5%85%A5%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%9A%84%E4%B8%BB%E5%AD%98%E5%82%A8%E5%99%A8%E6%8C%89%E5%AD%98%E5%82%A8%E5%99%A8%E7%9A%84%E6%8C%87%E4%BB%A4%E6%89%A7%E8%A1%8C%E7%9B%B4%E5%88%B0%E7%A8%8B%E5%BA%8F%E6%89%A7%E8%A1%8C%E7%BB%93%E6%9D%9F">概念：将<strong>指令</strong>以<strong>二进制码</strong>的形式实现输入计算机的<strong>主存储器</strong>，按存储器的指令执行，直到程序执行结束。</a></li>
<li><a href="#%E7%BB%93%E6%9E%84">结构：</a></li>
<li><a href="#%E7%89%B9%E7%82%B9">特点：</a></li>
</ul>
</li>
<li><a href="#%E7%8E%B0%E4%BB%A3%E8%AE%A1%E7%AE%97%E6%9C%BA">现代计算机</a><ul>
<li><a href="#%E7%BB%93%E6%9E%84-1">结构：</a></li>
<li><a href="#%E7%89%B9%E7%82%B9-1">特点：</a></li>
</ul>
</li>
<li><a href="#%E7%9F%A5%E8%AF%86%E6%A1%86%E6%9E%B6">知识框架</a></li>
</ul>
</li>
<li><a href="#%E5%90%84%E4%B8%AA%E7%A1%AC%E4%BB%B6%E7%9A%84%E5%B7%A5%E4%BD%9C%E5%8E%9F%E7%90%86">各个硬件的工作原理</a><ul>
<li><a href="#%E4%B8%BB%E5%AD%98%E5%82%A8%E5%99%A8">主存储器</a><ul>
<li><a href="#%E7%BB%84%E6%88%90">组成</a></li>
</ul>
</li>
<li><a href="#%E8%BF%90%E7%AE%97%E5%99%A8">运算器</a><ul>
<li><a href="#%E4%BD%9C%E7%94%A8">作用：</a></li>
<li><a href="#%E7%BB%84%E6%88%90-1">组成</a></li>
</ul>
</li>
<li><a href="#%E6%8E%A7%E5%88%B6%E5%99%A8">控制器</a><ul>
<li><a href="#%E7%BB%84%E6%88%90-2">组成</a></li>
</ul>
</li>
<li><a href="#%E5%B7%A5%E4%BD%9C%E8%BF%87%E7%A8%8B">工作过程</a></li>
<li><a href="#%E7%9F%A5%E8%AF%86%E6%A1%86%E6%9E%B6-1">知识框架</a></li>
</ul>
</li>
<li><a href="#%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E7%9A%84%E5%B1%82%E6%AC%A1%E7%BB%93%E6%9E%84">计算机系统的层次结构</a><ul>
<li><a href="#%E7%9F%A5%E8%AF%86%E6%A1%86%E6%9E%B6-2">知识框架</a></li>
</ul>
</li>
<li><a href="#%E8%AE%A1%E7%AE%97%E6%9C%BA%E6%80%A7%E8%83%BD%E6%8C%87%E6%A0%87">计算机性能指标</a><ul>
<li><a href="#%E5%AD%98%E5%82%A8%E5%99%A8%E6%80%A7%E8%83%BD%E6%8C%87%E6%A0%87">存储器性能指标</a><ul>
<li><a href="#%E6%80%BB%E5%AE%B9%E9%87%8F">总容量</a></li>
</ul>
</li>
<li><a href="#cpu%E6%80%A7%E8%83%BD%E6%8C%87%E6%A0%87">CPU性能指标</a><ul>
<li><a href="#cpu%E4%B8%BB%E9%A2%91cpu%E5%86%85%E6%95%B0%E5%AD%97%E8%84%89%E5%86%B2%E4%BF%A1%E5%8F%B7%E9%9C%87%E8%8D%A1%E7%9A%84%E9%A2%91%E7%8E%87">CPU主频：CPU内数字脉冲信号震荡的频率</a></li>
<li><a href="#cpi%E6%89%A7%E8%A1%8C%E4%B8%80%E6%9D%A1%E6%8C%87%E4%BB%A4%E6%89%80%E9%9C%80%E8%A6%81%E7%9A%84%E6%97%B6%E9%92%9F%E5%91%A8%E6%9C%9F%E6%95%B0%E4%B8%8D%E5%90%8C%E7%9A%84%E6%8C%87%E4%BB%A4cpi%E4%B8%8D%E5%90%8C">CPI：执行一条指令所需要的时钟周期数（不同的指令CPI不同）</a></li>
<li><a href="#%E8%80%97%E6%97%B6--cpi--cpu%E6%97%B6%E9%92%9F%E5%91%A8%E6%9C%9F">耗时 = CPI * CPU时钟周期</a></li>
<li><a href="#ips%E6%AF%8F%E7%A7%92%E6%89%A7%E8%A1%8C%E5%A4%9A%E5%B0%91%E6%9D%A1%E6%8C%87%E4%BB%A4">IPS：每秒执行多少条指令</a></li>
<li><a href="#flops%E6%AF%8F%E7%A7%92%E6%89%A7%E8%A1%8C%E5%A4%9A%E5%B0%91%E6%AC%A1%E6%B5%AE%E7%82%B9%E6%95%B0%E8%BF%90%E7%AE%97">FLOPS：每秒执行多少次浮点数运算</a></li>
</ul>
</li>
<li><a href="#%E7%B3%BB%E7%BB%9F%E6%95%B4%E4%BD%93%E6%80%A7%E8%83%BD%E6%8C%87%E6%A0%87">系统整体性能指标</a><ul>
<li><a href="#%E6%95%B0%E6%8D%AE%E9%80%9A%E8%B7%AF%E5%B8%A6%E5%AE%BD">数据通路带宽</a></li>
<li><a href="#%E5%90%9E%E5%90%90%E9%87%8F">吞吐量</a></li>
<li><a href="#%E5%93%8D%E5%BA%94%E6%97%B6%E9%97%B4">响应时间</a></li>
</ul>
</li>
<li><a href="#%E7%9F%A5%E8%AF%86%E6%A1%86%E6%9E%B6-3">知识框架</a></li>
</ul>
</li>
<li><a href="#%E8%BF%9B%E4%BD%8D%E8%AE%A1%E6%95%B0%E5%88%B6">进位计数制</a><ul>
<li><a href="#%E5%9F%BA%E6%95%B0">基数</a></li>
<li><a href="#%E4%BD%BF%E7%94%A8%E4%BA%8C%E8%BF%9B%E5%88%B6%E7%9A%84%E5%8E%9F%E5%9B%A0">使用二进制的原因</a></li>
<li><a href="#%E8%BF%9B%E5%88%B6%E8%BD%AC%E5%8C%96">进制转化</a></li>
<li><a href="#%E8%BF%9B%E5%88%B6%E4%B9%A6%E5%86%99%E6%96%B9%E5%BC%8F">进制书写方式</a></li>
<li><a href="#%E7%9C%9F%E5%80%BC%E5%92%8C%E6%9C%BA%E5%99%A8%E6%95%B0">真值和机器数</a></li>
<li><a href="#%E7%9F%A5%E8%AF%86%E6%A1%86%E6%9E%B6-4">知识框架</a></li>
</ul>
</li>
<li><a href="#bcd%E7%A0%81">BCD码</a><ul>
<li><a href="#8421">8421</a><ul>
<li><a href="#%E8%AE%A1%E7%AE%97%E6%9C%BA%E8%AE%A1%E7%AE%978421">计算机计算8421</a></li>
</ul>
</li>
<li><a href="#%E4%BD%99%E4%B8%89%E7%A0%81">余三码</a></li>
</ul>
</li>
<li><a href="#%E5%AD%97%E7%AC%A6%E4%B8%8E%E5%AD%97%E7%AC%A6%E4%B8%B2">字符与字符串</a><ul>
<li><a href="#ascii%E7%A0%81">ASCII码</a></li>
<li><a href="#%E5%AD%97%E7%AC%A6%E4%B8%B2">字符串</a></li>
</ul>
</li>
<li><a href="#%E5%A5%87%E5%81%B6%E6%A0%A1%E9%AA%8C%E7%A0%81">奇偶校验码</a><ul>
<li><a href="#%E7%A0%81%E5%AD%97%E8%8B%A5%E5%B9%B2%E4%BD%8D%E4%BB%A3%E7%A0%81%E7%BB%84%E6%88%90%E7%9A%84%E4%B8%80%E4%B8%AA%E5%AD%97">码字：若干位代码组成的一个字</a></li>
<li><a href="#%E7%A0%81%E8%B7%9D%E5%90%88%E6%B3%95%E7%A0%81%E5%AD%97%E4%B9%8B%E9%97%B4%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%9D%E7%A6%BB">码距：合法码字之间的最小距离</a></li>
<li><a href="#%E5%A5%87%E5%81%B6%E6%A0%A1%E9%AA%8C">奇偶校验</a><ul>
<li><a href="#%E7%A1%AC%E4%BB%B6%E5%AE%9E%E7%8E%B0">硬件实现</a></li>
</ul>
</li>
<li><a href="#%E7%9F%A5%E8%AF%86%E6%A1%86%E6%9E%B6-5">知识框架</a></li>
</ul>
</li>
<li><a href="#%E6%B5%B7%E6%98%8E%E6%A0%A1%E9%AA%8C%E7%A0%81">海明校验码</a><ul>
<li><a href="#%E8%AE%A1%E7%AE%97">计算</a></li>
<li><a href="#%E7%BA%A0%E9%94%99">纠错</a></li>
<li><a href="#%E8%A1%A5%E5%85%85">补充</a></li>
<li><a href="#%E7%9F%A5%E8%AF%86%E6%A1%86%E6%9E%B6-6">知识框架</a></li>
</ul>
</li>
<li><a href="#%E5%BE%AA%E7%8E%AF%E5%86%97%E4%BD%99%E6%A0%A1%E9%AA%8C%E7%A0%81crccyclic-redundany-check">循环冗余校验码(CRC,Cyclic Redundany Check)</a><ul>
<li><a href="#%E8%AE%A1%E7%AE%97crc">计算CRC</a></li>
<li><a href="#%E6%A0%A1%E9%AA%8C%E4%BD%8D%E8%AE%A1%E7%AE%97">校验位计算</a></li>
<li><a href="#%E6%A3%80%E9%94%99%E5%92%8C%E7%BA%A0%E9%94%99">检错和纠错</a></li>
<li><a href="#%E7%9F%A5%E8%AF%86%E6%A1%86%E6%9E%B6-7">知识框架</a></li>
</ul>
</li>
<li><a href="#%E5%AE%9A%E7%82%B9%E6%95%B0">定点数</a><ul>
<li><a href="#%E5%AE%9A%E4%B9%89%E5%B0%8F%E6%95%B0%E7%82%B9%E7%9A%84%E4%BD%8D%E7%BD%AE%E5%9B%BA%E5%AE%9A%E4%B8%8D%E5%8F%98-eg977007">定义：小数点的位置固定不变 eg：977.007</a></li>
<li><a href="#%E5%8E%9F%E7%A0%81">原码</a><ul>
<li><a href="#%E8%A1%A8%E7%A4%BA%E8%8C%83%E5%9B%B4">表示范围：</a></li>
<li><a href="#%E4%B9%A6%E5%86%99">书写</a></li>
</ul>
</li>
<li><a href="#%E5%8F%8D%E7%A0%81">反码</a><ul>
<li><a href="#%E4%B9%A6%E5%86%99-1">书写</a></li>
</ul>
</li>
<li><a href="#%E8%A1%A5%E7%A0%81">补码</a><ul>
<li><a href="#%E4%B9%A6%E5%86%99-2">书写</a></li>
<li><a href="#%E8%8C%83%E5%9B%B4">范围</a></li>
</ul>
</li>
<li><a href="#%E7%A7%BB%E7%A0%81">移码</a><ul>
<li><a href="#%E4%B9%A6%E5%86%99%E5%B0%86%E8%A1%A5%E7%A0%81%E7%AC%A6%E5%8F%B7%E4%BD%8D%E5%8F%96%E5%8F%8D">书写：将补码符号位取反</a></li>
</ul>
</li>
<li><a href="#%E7%9F%A5%E8%AF%86%E6%A1%86%E6%9E%B6-8">知识框架</a></li>
</ul>
</li>
<li><a href="#%E5%90%84%E7%A7%8D%E7%A0%81%E7%9A%84%E4%BD%9C%E7%94%A8">各种码的作用</a><ul>
<li><a href="#%E5%8E%9F%E7%A0%81%E7%9A%84%E5%8A%A0%E5%87%8F">原码的加减</a><ul>
<li><a href="#%E6%A8%A1%E8%BF%90%E7%AE%97">模运算</a></li>
<li><a href="#%E5%8A%A0%E6%B3%95%E8%BF%90%E7%AE%97">加法运算</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#%E7%A7%BB%E4%BD%8D%E8%BF%90%E7%AE%97">移位运算</a><ul>
<li><a href="#%E7%AE%97%E6%95%B0%E7%A7%BB%E4%BD%8D%E6%AF%8F%E7%A7%BB%E5%8A%A8%E4%B8%80%E4%BD%8D%E7%9B%B8%E5%BD%93%E4%BA%8E%E6%88%962%E5%AF%B9%E4%BA%8C%E8%BF%9B%E5%88%B6">算数移位：每移动一位相当于*或/2（对二进制）</a></li>
<li><a href="#%E9%80%BB%E8%BE%91%E7%A7%BB%E4%BD%8D">逻辑移位</a></li>
<li><a href="#%E5%BE%AA%E7%8E%AF%E7%A7%BB%E4%BD%8D">循环移位</a></li>
<li><a href="#%E7%9F%A5%E8%AF%86%E6%A1%86%E6%9E%B6-9">知识框架</a></li>
</ul>
</li>
<li><a href="#%E5%8A%A0%E5%87%8F%E8%BF%90%E7%AE%97">加减运算</a><ul>
<li><a href="#%E6%BA%A2%E5%87%BA">溢出</a><ul>
<li><a href="#%E5%88%A4%E6%96%AD">判断：</a></li>
</ul>
</li>
<li><a href="#%E7%AC%A6%E5%8F%B7%E6%89%A9%E5%B1%95">符号扩展</a></li>
</ul>
</li>
<li><a href="#%E5%AE%9A%E7%82%B9%E6%95%B0%E7%9A%84%E4%B9%98%E6%B3%95">定点数的乘法</a><ul>
<li><a href="#%E5%8E%9F%E7%A0%81%E4%B9%98%E6%B3%95">原码乘法</a></li>
<li><a href="#%E8%A1%A5%E7%A0%81%E7%9A%84%E4%B9%98%E6%B3%95">补码的乘法</a><ul>
<li><a href="#%E5%AF%B9%E6%AF%94%E5%8E%9F%E7%A0%81%E5%8A%A0%E6%B3%95">对比原码加法</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#%E5%AE%9A%E7%82%B9%E6%95%B0%E7%9A%84%E9%99%A4%E6%B3%95">定点数的除法</a><ul>
<li><a href="#%E5%8E%9F%E7%A0%81%E7%9A%84%E9%99%A4%E6%B3%95">原码的除法</a><ul>
<li><a href="#%E6%81%A2%E5%A4%8D%E4%BD%99%E6%95%B0%E6%B3%95">恢复余数法</a></li>
<li><a href="#%E5%8A%A0%E5%87%8F%E4%BA%A4%E6%9B%BF%E6%B3%95">加减交替法</a></li>
</ul>
</li>
<li><a href="#%E8%A1%A5%E7%A0%81%E7%9A%84%E9%99%A4%E6%B3%95">补码的除法</a></li>
<li><a href="#%E5%8A%A0%E5%87%8F%E4%BA%A4%E6%9B%BF%E6%B3%95%E6%9C%AB%E5%B0%BE%E6%81%92%E5%95%861">加减交替法（末尾恒商1）</a></li>
<li><a href="#%E5%8E%9F%E7%A0%81%E5%92%8C%E8%A1%A5%E7%A0%81%E7%9A%84%E5%8A%A0%E5%87%8F%E4%BA%A4%E6%9B%BF%E6%B3%95%E7%9A%84%E5%AF%B9%E6%AF%94">原码和补码的加减交替法的对比</a></li>
</ul>
</li>
<li><a href="#%E5%BC%BA%E5%88%B6%E7%B1%BB%E5%9E%8B%E8%BD%AC%E6%8D%A2">强制类型转换</a></li>
<li><a href="#%E6%95%B0%E6%8D%AE%E5%AD%98%E5%82%A8">数据存储</a><ul>
<li><a href="#%E5%A4%A7%E5%B0%8F%E7%AB%AF%E6%A8%A1%E5%BC%8F">大小端模式</a></li>
<li><a href="#%E8%BE%B9%E7%95%8C%E5%AF%B9%E9%BD%90%E7%A9%BA%E9%97%B4%E6%8D%A2%E6%97%B6%E9%97%B4">边界对齐（空间换时间）</a></li>
</ul>
</li>
<li><a href="#%E6%B5%AE%E7%82%B9%E6%95%B0%E7%9A%84%E8%A1%A8%E7%A4%BA">浮点数的表示</a><ul>
<li><a href="#%E8%A7%84%E6%A0%BC%E5%8C%96">规格化</a></li>
</ul>
</li>
<li><a href="#%E6%B5%AE%E7%82%B9%E6%95%B0%E6%A0%87%E5%87%86">浮点数标准</a><ul>
<li><a href="#ieee-754">IEEE 754</a><ul>
<li><a href="#%E7%BB%9D%E5%AF%B9%E5%80%BC%E8%8C%83%E5%9B%B4">绝对值范围</a></li>
<li><a href="#%E9%98%B6%E7%A0%81%E7%9A%84%E7%89%B9%E6%AE%8A%E6%83%85%E5%86%B5">阶码的特殊情况</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#%E6%B5%AE%E7%82%B9%E6%95%B0%E7%9A%84%E8%BF%90%E7%AE%97">浮点数的运算</a><ul>
<li><a href="#%E6%96%B9%E6%B3%95">方法</a></li>
<li><a href="#%E8%88%8D%E5%85%A5">舍入</a></li>
<li><a href="#%E5%BC%BA%E5%88%B6%E7%B1%BB%E5%9E%8B%E8%BD%AC%E6%8D%A2-1">强制类型转换</a></li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- /TOC -->
<h1 id="计算机组成原理"><a href="#计算机组成原理" class="headerlink" title="计算机组成原理"></a>计算机组成原理</h1><h2 id="计算机系统"><a href="#计算机系统" class="headerlink" title="计算机系统"></a>计算机系统</h2><h3 id="计算机系统-硬件-软件-软件和硬件等效"><a href="#计算机系统-硬件-软件-软件和硬件等效" class="headerlink" title="计算机系统 = 硬件 + 软件 (软件和硬件等效)"></a>计算机系统 = 硬件 + 软件 (软件和硬件等效)</h3><h4 id="软件"><a href="#软件" class="headerlink" title="软件"></a>软件</h4><ol>
<li>系统软件：管理整个计算机系统<br>eg：操作系统，数据库管理系统，标准程序库，网络软件，语言处理程序，服务程序。</li>
<li>应用软件：按任务需要完成各种指令</li>
</ol>
<hr>
<h2 id="计算机硬件的基本组成"><a href="#计算机硬件的基本组成" class="headerlink" title="计算机硬件的基本组成"></a>计算机硬件的基本组成</h2><h3 id="冯诺依曼机"><a href="#冯诺依曼机" class="headerlink" title="冯诺依曼机"></a>冯诺依曼机</h3><h4 id="概念：将指令以二进制码的形式实现输入计算机的主存储器，按存储器的指令执行，直到程序执行结束。"><a href="#概念：将指令以二进制码的形式实现输入计算机的主存储器，按存储器的指令执行，直到程序执行结束。" class="headerlink" title="概念：将指令以二进制码的形式实现输入计算机的主存储器，按存储器的指令执行，直到程序执行结束。"></a>概念：将<strong>指令</strong>以<strong>二进制码</strong>的形式实现输入计算机的<strong>主存储器</strong>，按存储器的指令执行，直到程序执行结束。</h4><h4 id="结构："><a href="#结构：" class="headerlink" title="结构："></a>结构：</h4><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-20_10-20-16.png"></p>
<h4 id="特点："><a href="#特点：" class="headerlink" title="特点："></a>特点：</h4><ol>
<li>五大部件组成</li>
<li>指令和数据以同等地位存于存储器，可按地址寻访。</li>
<li>指令和数据以二进制表示</li>
<li>指令由操作码(+,-,*,%)和地址码(需要操作的数据的地址)组成</li>
<li>存储程序</li>
<li>以运算器为中心</li>
</ol>
<h3 id="现代计算机"><a href="#现代计算机" class="headerlink" title="现代计算机"></a>现代计算机</h3><h4 id="结构：-1"><a href="#结构：-1" class="headerlink" title="结构："></a>结构：</h4><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-20_10-30-27.png"><br>CPU = 运算器 + 控制器<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-20_10-34-42.png"></p>
<h4 id="特点：-1"><a href="#特点：-1" class="headerlink" title="特点："></a>特点：</h4><ol>
<li>以存储器为中心<h3 id="知识框架"><a href="#知识框架" class="headerlink" title="知识框架"></a>知识框架</h3><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-20_10-36-31.png"></li>
</ol>
<hr>
<h2 id="各个硬件的工作原理"><a href="#各个硬件的工作原理" class="headerlink" title="各个硬件的工作原理"></a>各个硬件的工作原理</h2><h3 id="主存储器"><a href="#主存储器" class="headerlink" title="主存储器"></a>主存储器</h3><h4 id="组成"><a href="#组成" class="headerlink" title="组成"></a>组成</h4><ol>
<li>存储体<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-20_11-05-19.png"><br>存储字（word）：存储单元中二进制代码组合<br>存储字长：存储单元中二进制码的位数</li>
<li>MAR（Memory Address Register 地址寄存器）<br>MAR位数反应存储单元个数<br>MAR4位 -&gt; 2^4个存储单元</li>
<li>MDR（Memory Data Register 数据寄存器）<br>MDR位数 = 存储字长<br>MDR16位每个存储单元可存放16 bit<br>即一个字（word）= 16 bit<br>1B = 1字节 1b = 1位</li>
</ol>
<h3 id="运算器"><a href="#运算器" class="headerlink" title="运算器"></a>运算器</h3><h4 id="作用："><a href="#作用：" class="headerlink" title="作用："></a>作用：</h4><p>实现算术运算，逻辑运算  </p>
<h4 id="组成-1"><a href="#组成-1" class="headerlink" title="组成"></a>组成</h4><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-20_11-16-53.png"></p>
<ol>
<li>ACC（accumulator）：累加器，存放操作数和运算结果</li>
<li>MQ：乘商寄存器，乘除运算时存放操作数和运算结果</li>
<li>X：通用操作数寄存器，存放操作数</li>
<li>ALU：算术逻辑单元，实现算术运算和逻辑运算 </li>
</ol>
<h3 id="控制器"><a href="#控制器" class="headerlink" title="控制器"></a>控制器</h3><h4 id="组成-2"><a href="#组成-2" class="headerlink" title="组成"></a>组成</h4><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-20_11-18-28.png"></p>
<ol>
<li>CU: 控制单元，分析指令,发出控制信号</li>
<li>IR：指令寄存器，存放当前执行指令</li>
<li>PC：程序计数器，存放下一条指令地址，自动+1  </li>
</ol>
<p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-20_11-23-50.png"></p>
<h3 id="工作过程"><a href="#工作过程" class="headerlink" title="工作过程"></a>工作过程</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-20_13-46-34.png"></p>
<h3 id="知识框架-1"><a href="#知识框架-1" class="headerlink" title="知识框架"></a>知识框架</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-20_13-49-19.png"></p>
<hr>
<h2 id="计算机系统的层次结构"><a href="#计算机系统的层次结构" class="headerlink" title="计算机系统的层次结构"></a>计算机系统的层次结构</h2><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-20_13-58-41.png"></p>
<h3 id="知识框架-2"><a href="#知识框架-2" class="headerlink" title="知识框架"></a>知识框架</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-20_14-06-04.png"></p>
<hr>
<h2 id="计算机性能指标"><a href="#计算机性能指标" class="headerlink" title="计算机性能指标"></a>计算机性能指标</h2><h3 id="存储器性能指标"><a href="#存储器性能指标" class="headerlink" title="存储器性能指标"></a>存储器性能指标</h3><h4 id="总容量"><a href="#总容量" class="headerlink" title="总容量"></a>总容量</h4><p>总容量 = 存储单元个数 * 存储字长/8 Byte<br>eg：MAR 32位，MDR 8位<br>    总容量 = 2^32*8 = 4GB</p>
<h3 id="CPU性能指标"><a href="#CPU性能指标" class="headerlink" title="CPU性能指标"></a>CPU性能指标</h3><h4 id="CPU主频：CPU内数字脉冲信号震荡的频率"><a href="#CPU主频：CPU内数字脉冲信号震荡的频率" class="headerlink" title="CPU主频：CPU内数字脉冲信号震荡的频率"></a>CPU主频：CPU内数字脉冲信号震荡的频率</h4><p>CPU主频 = 1 / CPU时钟周期<br>单位:GHz(2^30Hz)</p>
<h4 id="CPI：执行一条指令所需要的时钟周期数（不同的指令CPI不同）"><a href="#CPI：执行一条指令所需要的时钟周期数（不同的指令CPI不同）" class="headerlink" title="CPI：执行一条指令所需要的时钟周期数（不同的指令CPI不同）"></a>CPI：执行一条指令所需要的时钟周期数（不同的指令CPI不同）</h4><h4 id="耗时-CPI-CPU时钟周期"><a href="#耗时-CPI-CPU时钟周期" class="headerlink" title="耗时 = CPI * CPU时钟周期"></a>耗时 = CPI * CPU时钟周期</h4><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-20_14-22-25.png"></p>
<h4 id="IPS：每秒执行多少条指令"><a href="#IPS：每秒执行多少条指令" class="headerlink" title="IPS：每秒执行多少条指令"></a>IPS：每秒执行多少条指令</h4><p>IPS = 主频 / 平均CPI</p>
<h4 id="FLOPS：每秒执行多少次浮点数运算"><a href="#FLOPS：每秒执行多少次浮点数运算" class="headerlink" title="FLOPS：每秒执行多少次浮点数运算"></a>FLOPS：每秒执行多少次浮点数运算</h4><h3 id="系统整体性能指标"><a href="#系统整体性能指标" class="headerlink" title="系统整体性能指标"></a>系统整体性能指标</h3><h4 id="数据通路带宽"><a href="#数据通路带宽" class="headerlink" title="数据通路带宽"></a>数据通路带宽</h4><p>数据总线一次所能并行传输信息的位数（各个硬件通过数据总线传输数据）</p>
<h4 id="吞吐量"><a href="#吞吐量" class="headerlink" title="吞吐量"></a>吞吐量</h4><p>系统在单位时间内处理请求的数量。</p>
<h4 id="响应时间"><a href="#响应时间" class="headerlink" title="响应时间"></a>响应时间</h4><p>从用户向计算机发送一个请求，到系统对请求做出响应所用时间。</p>
<h3 id="知识框架-3"><a href="#知识框架-3" class="headerlink" title="知识框架"></a>知识框架</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-20_14-36-49.png"></p>
<hr>
<h2 id="进位计数制"><a href="#进位计数制" class="headerlink" title="进位计数制"></a>进位计数制</h2><h3 id="基数"><a href="#基数" class="headerlink" title="基数"></a>基数</h3><p>n进制数的基数个数为n，eg：10进制0，1，2，3，4，5，6，7，8，9</p>
<h3 id="使用二进制的原因"><a href="#使用二进制的原因" class="headerlink" title="使用二进制的原因"></a>使用二进制的原因</h3><ol>
<li>可以使用两个稳定状态的物理器件表示（eg：高低电频）</li>
<li>0和1对应逻辑值的真假，方便逻辑运算</li>
<li>方便使用逻辑门电路实现算术运算</li>
</ol>
<h3 id="进制转化"><a href="#进制转化" class="headerlink" title="进制转化"></a>进制转化</h3><ol>
<li>2进制-&gt;8进制（每三位一组）<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-26_09-16-36.png"></li>
<li>2进制-&gt;16进制（每四位一组）<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-26_09-17-45.png"></li>
<li>10进制-&gt;其他  </li>
</ol>
<p>eg：75.3<br>整数部分<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-26_09-22-32.png"><br>小数部分<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-26_09-24-33.png"></p>
<h3 id="进制书写方式"><a href="#进制书写方式" class="headerlink" title="进制书写方式"></a>进制书写方式</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-26_09-19-44.png"></p>
<h3 id="真值和机器数"><a href="#真值和机器数" class="headerlink" title="真值和机器数"></a>真值和机器数</h3><p>真值：符合人类习惯的数字（+15，-8）<br>机器数：数字实际存到机器里的形式（0 1111，1 100）</p>
<h3 id="知识框架-4"><a href="#知识框架-4" class="headerlink" title="知识框架"></a>知识框架</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-26_09-29-02.png"></p>
<hr>
<h2 id="BCD码"><a href="#BCD码" class="headerlink" title="BCD码"></a>BCD码</h2><h3 id="8421"><a href="#8421" class="headerlink" title="8421"></a>8421</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-26_10-19-38.png"></p>
<p>eg：985-&gt;1001 1000 0101</p>
<h4 id="计算机计算8421"><a href="#计算机计算8421" class="headerlink" title="计算机计算8421"></a>计算机计算8421</h4><p>5 + 8 = 0101 + 1000 = 1101（不在映射关系内）<br>将 1101 + 0110 = 1 0011 -&gt; 0001 0011 </p>
<h3 id="余三码"><a href="#余三码" class="headerlink" title="余三码"></a>余三码</h3><p>8421码 + 0011<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-26_10-24-23.png"></p>
<hr>
<h2 id="字符与字符串"><a href="#字符与字符串" class="headerlink" title="字符与字符串"></a>字符与字符串</h2><h3 id="ASCII码"><a href="#ASCII码" class="headerlink" title="ASCII码"></a>ASCII码</h3><blockquote>
<p>已知‘A’ 的ASCII码值为65，字符‘H’ 存放在某存储单元M中，求M中存放的内容。</p>
<blockquote>
<p>H的ASCII码为72，存放的内容为8位二进制0100 1000</p>
</blockquote>
</blockquote>
<h3 id="字符串"><a href="#字符串" class="headerlink" title="字符串"></a>字符串</h3><ol>
<li>大端模式：最高有效字节存放在低地址</li>
<li>小端模式：最高有效字节存放在高地址</li>
</ol>
<hr>
<h2 id="奇偶校验码"><a href="#奇偶校验码" class="headerlink" title="奇偶校验码"></a>奇偶校验码</h2><h3 id="码字：若干位代码组成的一个字"><a href="#码字：若干位代码组成的一个字" class="headerlink" title="码字：若干位代码组成的一个字"></a>码字：若干位代码组成的一个字</h3><h3 id="码距：合法码字之间的最小距离"><a href="#码距：合法码字之间的最小距离" class="headerlink" title="码距：合法码字之间的最小距离"></a>码距：合法码字之间的最小距离</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-26_10-51-59.png"><br>d = 1,无纠错能力，d = 2，有纠错能力，d &gt;= 3，有纠错检错能力</p>
<h3 id="奇偶校验"><a href="#奇偶校验" class="headerlink" title="奇偶校验"></a>奇偶校验</h3><ol>
<li>奇校验码：1的个数位奇数</li>
<li>偶校验码：1的个数位偶数  </li>
</ol>
<p>eg：1001101<br>奇校验码：<strong>1</strong>1001101<br>偶校验码：<strong>0</strong>1001101</p>
<h4 id="硬件实现"><a href="#硬件实现" class="headerlink" title="硬件实现"></a>硬件实现</h4><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-26_11-46-22.png"></p>
<h3 id="知识框架-5"><a href="#知识框架-5" class="headerlink" title="知识框架"></a>知识框架</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-26_11-47-27.png"></p>
<hr>
<h2 id="海明校验码"><a href="#海明校验码" class="headerlink" title="海明校验码"></a>海明校验码</h2><p>信息位 n, 校验位k<br>2^k(校验位的状态) &gt;= n+k+1(信息位+校验位+一种正确状态)</p>
<h3 id="计算"><a href="#计算" class="headerlink" title="计算"></a>计算</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-27_10-17-15.png"><br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-27_10-18-10.png"></p>
<h3 id="纠错"><a href="#纠错" class="headerlink" title="纠错"></a>纠错</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-27_10-19-34.png"></p>
<h3 id="补充"><a href="#补充" class="headerlink" title="补充"></a>补充</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-27_10-23-04.png"></p>
<h3 id="知识框架-6"><a href="#知识框架-6" class="headerlink" title="知识框架"></a>知识框架</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-27_10-25-09.png"></p>
<hr>
<h2 id="循环冗余校验码-CRC-Cyclic-Redundany-Check"><a href="#循环冗余校验码-CRC-Cyclic-Redundany-Check" class="headerlink" title="循环冗余校验码(CRC,Cyclic Redundany Check)"></a>循环冗余校验码(CRC,Cyclic Redundany Check)</h2><h3 id="计算CRC"><a href="#计算CRC" class="headerlink" title="计算CRC"></a>计算CRC</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-27_10-36-03.png"></p>
<h3 id="校验位计算"><a href="#校验位计算" class="headerlink" title="校验位计算"></a>校验位计算</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-27_10-33-59.png"></p>
<h3 id="检错和纠错"><a href="#检错和纠错" class="headerlink" title="检错和纠错"></a>检错和纠错</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-27_10-39-47.png"><br>也可能是第9位出错，对于确定多项式，出错位数是一定的与信息位无关<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-27_10-41-38.png"><br>K个信息位，R个校验位，若2^R &gt;= K+R+1，可纠正一位错误</p>
<h3 id="知识框架-7"><a href="#知识框架-7" class="headerlink" title="知识框架"></a>知识框架</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-27_10-52-14.png"></p>
<hr>
<h2 id="定点数"><a href="#定点数" class="headerlink" title="定点数"></a>定点数</h2><h3 id="定义：小数点的位置固定不变-eg：977-007"><a href="#定义：小数点的位置固定不变-eg：977-007" class="headerlink" title="定义：小数点的位置固定不变 eg：977.007"></a>定义：小数点的位置固定不变 eg：977.007</h3><p>浮点数：小数点的位置不固定 eg：9.97007*10^2</p>
<ol>
<li>无符号数<br>只有无符号整数没有无符号小数<br>n位无符号数<strong>表示范围</strong>：0 ~ 2^n - 1</li>
<li>有符号数（包括原码，反码，补码）<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-27_11-03-28.png"></li>
</ol>
<h3 id="原码"><a href="#原码" class="headerlink" title="原码"></a>原码</h3><p>符号位“0/1”，对应“正/负”</p>
<h4 id="表示范围："><a href="#表示范围：" class="headerlink" title="表示范围："></a>表示范围：</h4><ol>
<li>整数：<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-27_11-18-14.png">  </li>
<li>小数<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-27_11-17-03.png"></li>
</ol>
<p>0的真值有+0和-0  </p>
<h4 id="书写"><a href="#书写" class="headerlink" title="书写"></a>书写</h4><p>机器字长为8时<br>+19D -&gt; <strong>0</strong>,0010011<br>-19D -&gt; <strong>1</strong>,0010011<br>+0.75 -&gt; <strong>0</strong>.1100000<br> -0.75 -&gt; <strong>1</strong>.1100000  </p>
<h3 id="反码"><a href="#反码" class="headerlink" title="反码"></a>反码</h3><h4 id="书写-1"><a href="#书写-1" class="headerlink" title="书写"></a>书写</h4><p>符号位为<strong>0</strong>，与原码相同<br>符号位为<strong>1</strong>，原码数值位取反<br>+19D -&gt; <strong>0</strong>,0010011<br>-19D -&gt; <strong>1</strong>,1101100<br>+0.75 -&gt; <strong>0</strong>.1100000<br>-0.75 -&gt; <strong>1</strong>.0011111   </p>
<h3 id="补码"><a href="#补码" class="headerlink" title="补码"></a>补码</h3><h4 id="书写-2"><a href="#书写-2" class="headerlink" title="书写"></a>书写</h4><p>符号位为<strong>0</strong>，与原码相同<br>符号位为<strong>1</strong>，反码末位 + 1<br>+19D -&gt; <strong>0</strong>,0010011<br>-19D -&gt; <strong>1</strong>,1101101<br>+0.75 -&gt; <strong>0</strong>.1100000<br>-0.75 -&gt; <strong>1</strong>.0100000<br>0的补码：00000000</p>
<h4 id="范围"><a href="#范围" class="headerlink" title="范围"></a>范围</h4><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-27_13-57-11.png"></p>
<h3 id="移码"><a href="#移码" class="headerlink" title="移码"></a>移码</h3><h4 id="书写：将补码符号位取反"><a href="#书写：将补码符号位取反" class="headerlink" title="书写：将补码符号位取反"></a>书写：将补码符号位取反</h4><p>+19D -&gt; <strong>1</strong>,0010011<br>-19D -&gt; <strong>0</strong>,1101101<br>+0.75 -&gt; <strong>1</strong>.1100000<br>-0.75 -&gt; <strong>0</strong>.0100000   </p>
<h3 id="知识框架-8"><a href="#知识框架-8" class="headerlink" title="知识框架"></a>知识框架</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-27_14-04-25.png"></p>
<hr>
<h2 id="各种码的作用"><a href="#各种码的作用" class="headerlink" title="各种码的作用"></a>各种码的作用</h2><h3 id="原码的加减"><a href="#原码的加减" class="headerlink" title="原码的加减"></a>原码的加减</h3><h4 id="模运算"><a href="#模运算" class="headerlink" title="模运算"></a>模运算</h4><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-29_08-14-17.png"><br>(-3) mod 12 = 9<br>9 mod 12 = 9<br>-3和9的绝对值之和为12，则称他们互为补数</p>
<h4 id="加法运算"><a href="#加法运算" class="headerlink" title="加法运算"></a>加法运算</h4><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-29_08-19-34.png"></p>
<hr>
<h2 id="移位运算"><a href="#移位运算" class="headerlink" title="移位运算"></a>移位运算</h2><h3 id="算数移位：每移动一位相当于-或-2（对二进制）"><a href="#算数移位：每移动一位相当于-或-2（对二进制）" class="headerlink" title="算数移位：每移动一位相当于*或/2（对二进制）"></a>算数移位：每移动一位相当于*或/2（对二进制）</h3><p>原码：<strong>1</strong>0010100. -20D<br>左移：<strong>1</strong>0001010.0 -10D<br>左移：<strong>1</strong>0000010.100 -<strong>2</strong>D<br>右移：<strong>1</strong>0101000.  -40D<br><strong>如果舍弃的位数不是0而是1，则会损失精度</strong><br><strong>负数的反码移位补1</strong><br><strong>负数补码，最右边的1，右边同反码，左边同原码</strong><br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-29_08-37-54.png"></p>
<h3 id="逻辑移位"><a href="#逻辑移位" class="headerlink" title="逻辑移位"></a>逻辑移位</h3><p>右移：高位补0，低位舍弃<br>左移：低位补0，高位舍弃</p>
<h3 id="循环移位"><a href="#循环移位" class="headerlink" title="循环移位"></a>循环移位</h3><p>左移：10110101 -&gt; 0101101011</p>
<h3 id="知识框架-9"><a href="#知识框架-9" class="headerlink" title="知识框架"></a>知识框架</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-29_08-45-32.png"></p>
<hr>
<h2 id="加减运算"><a href="#加减运算" class="headerlink" title="加减运算"></a>加减运算</h2><h3 id="溢出"><a href="#溢出" class="headerlink" title="溢出"></a>溢出</h3><p>结果&gt;127上溢，正正得负<br>结果&lt;-128下溢，负负得正</p>
<h4 id="判断："><a href="#判断：" class="headerlink" title="判断："></a>判断：</h4><ol>
<li>a + b = c符号分别为ABC<br>V = AB(!C)+(!A)(!B)C<br>V = 0，无溢出，V = 1，有溢出 </li>
<li>双符号位<br>正数00，负数11<br>01上溢，10下溢<h3 id="符号扩展"><a href="#符号扩展" class="headerlink" title="符号扩展"></a>符号扩展</h3><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-29_09-10-28.png"></li>
</ol>
<hr>
<h2 id="定点数的乘法"><a href="#定点数的乘法" class="headerlink" title="定点数的乘法"></a>定点数的乘法</h2><h3 id="原码乘法"><a href="#原码乘法" class="headerlink" title="原码乘法"></a>原码乘法</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-29_09-22-37.png"><br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-04-29_09-24-21.png"></p>
<h3 id="补码的乘法"><a href="#补码的乘法" class="headerlink" title="补码的乘法"></a>补码的乘法</h3><h4 id="对比原码加法"><a href="#对比原码加法" class="headerlink" title="对比原码加法"></a>对比原码加法</h4><ol>
<li>比原码乘法最后多一次加法  </li>
<li>辅助位 - MQ“最低位” = 1时（ACC）+ [x]补<br>辅助位 - MQ“最低位” = 0时（ACC）+<br>辅助位 - MQ“最低位” = 1时（ACC）+ [-x]补  </li>
<li>每次进行补码的<strong>算数右移</strong></li>
<li>符号位参与运算</li>
<li>乘数是单符号位，被乘数是双符号位（因为乘数有一位被辅助位占据)</li>
</ol>
<p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-01_10-15-15.png"><br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-01_10-20-06.png"></p>
<hr>
<h2 id="定点数的除法"><a href="#定点数的除法" class="headerlink" title="定点数的除法"></a>定点数的除法</h2><h3 id="原码的除法"><a href="#原码的除法" class="headerlink" title="原码的除法"></a>原码的除法</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-01_10-28-56.png"></p>
<h4 id="恢复余数法"><a href="#恢复余数法" class="headerlink" title="恢复余数法"></a>恢复余数法</h4><p>计算机只能先商1，通过对于被除数的符号位判断，为1则改商为0，为0则不改变<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-01_10-42-33.png"><br>将ACC内的1110加上除数的补码恢复成原来的被除数01011<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-01_10-46-43.png"><br>逻辑左移<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-01_10-50-42.png"><br>结果<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-01_10-52-59.png"><br>手算<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-01_10-58-38.png"> </p>
<h4 id="加减交替法"><a href="#加减交替法" class="headerlink" title="加减交替法"></a>加减交替法</h4><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-01_11-03-44.png"><br>如果最后算出余数为负数则要加上除数的补码来恢复余数，这里是0.0111不用恢复</p>
<h3 id="补码的除法"><a href="#补码的除法" class="headerlink" title="补码的除法"></a>补码的除法</h3><h3 id="加减交替法（末尾恒商1）"><a href="#加减交替法（末尾恒商1）" class="headerlink" title="加减交替法（末尾恒商1）"></a>加减交替法（末尾恒商1）</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-02_11-24-08.png"> </p>
<h3 id="原码和补码的加减交替法的对比"><a href="#原码和补码的加减交替法的对比" class="headerlink" title="原码和补码的加减交替法的对比"></a>原码和补码的加减交替法的对比</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-02_11-26-08.png"> </p>
<hr>
<h2 id="强制类型转换"><a href="#强制类型转换" class="headerlink" title="强制类型转换"></a>强制类型转换</h2><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-02_11-33-35.png"> </p>
<hr>
<h2 id="数据存储"><a href="#数据存储" class="headerlink" title="数据存储"></a>数据存储</h2><h3 id="大小端模式"><a href="#大小端模式" class="headerlink" title="大小端模式"></a>大小端模式</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-02_11-39-20.png"></p>
<h3 id="边界对齐（空间换时间）"><a href="#边界对齐（空间换时间）" class="headerlink" title="边界对齐（空间换时间）"></a>边界对齐（空间换时间）</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-02_11-44-45.png"> </p>
<hr>
<h2 id="浮点数的表示"><a href="#浮点数的表示" class="headerlink" title="浮点数的表示"></a>浮点数的表示</h2><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-02_12-08-38.png"> </p>
<h3 id="规格化"><a href="#规格化" class="headerlink" title="规格化"></a>规格化</h3><p>尾数的数值部分（不算符号位）最高位必须是1<br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-02_12-21-50.png"> </p>
<h2 id="浮点数标准"><a href="#浮点数标准" class="headerlink" title="浮点数标准"></a>浮点数标准</h2><h3 id="IEEE-754"><a href="#IEEE-754" class="headerlink" title="IEEE 754"></a>IEEE 754</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-04_10-07-40.png"><br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-04_10-12-08.png"><br><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-04_10-16-33.png"></p>
<h4 id="绝对值范围"><a href="#绝对值范围" class="headerlink" title="绝对值范围"></a>绝对值范围</h4><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-04_10-19-26.png"></p>
<h4 id="阶码的特殊情况"><a href="#阶码的特殊情况" class="headerlink" title="阶码的特殊情况"></a>阶码的特殊情况</h4><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-04_10-25-54.png"></p>
<hr>
<h2 id="浮点数的运算"><a href="#浮点数的运算" class="headerlink" title="浮点数的运算"></a>浮点数的运算</h2><h3 id="方法"><a href="#方法" class="headerlink" title="方法"></a>方法</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-04_10-53-40.png"></p>
<h3 id="舍入"><a href="#舍入" class="headerlink" title="舍入"></a>舍入</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-04_10-55-15.png"></p>
<h3 id="强制类型转换-1"><a href="#强制类型转换-1" class="headerlink" title="强制类型转换"></a>强制类型转换</h3><p><img src="/2021/05/04/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/Snipaste_2021-05-04_11-03-11.png"></p>

      

      
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